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THE  matters  collected  in  this  Syllabus  will  be  found  in  those  of  my 
writings*  of  which  the  titles  follow: — I.  On  the  Structure  of  the 
Syllogism  ....  (Cambridge  Transactions,  vol.  viii,  part  3,  1847). 
II.  Formal  Logic,  or  the  Calculus  of  Inference,  necessary  and 
probable  (London,  Taylor  and  Walton,  1847,  8vo.).  III.  On  the 
Symbols  of  Logic,  the  Theory  of  the  Syllogism,  and  in  particular  of 
the  Copula,  ....  (Cambridge  Transactions,  vol.  ix,  part  1,  1850). 
IV.  On  the  Syllogism,  No.  iii,  and  on  Logic  in  general  (Cambridge 
Transactions,  vol.  x,  part  1,  1858).  Of  these  works  the  formulae 
and  notation  of  the  first  are  entirely  superseded ;  the  notation  only  of 
the  second  (the  Formal  Logic]  may  be  advantageously  replaced  (see 
§  24)  by  that  of  the  third  and  fourth  and  of  the  present  tract.  There 
is  very  little  in  the  first  three  writings  on  which  my  opinion  has 
varied ;  but  of  all  three  it  is  to  be  said  that  they  are  entirely  based 
on  what  I  now  call  the  arithmetical  view  6f  the  proposition  and 
syllogism  (§  8,  173,  174),  extending  this  term  not  merely  to  the 
numerically  definite  syllogism,  but  to  the  ordinary  form,  to  my  own 
extension  of  it,  and  to  Sir  W.  Hamilton's  departure  from  it. 

*  The  writings  which  oppose  any  of  my  views  at  length  are  the  following,  so  far 
as  my  memory  serves.  I.  A  letter  to  Augustus  De  Morgan,  Esq.  .  .  .  on  his  claim  to 
an  independent  rediscovery  of  a  new  principle  in  the  theory  of  syllogism,  from  Sir 

William  Hamilton,  Bart London  and  Edinburgh,  Longman  and  Co.,  Mach- 

lachlan  and  Co.,  1847,  8vo.  II.  Review  of  my  Formal  Logic,  signed  J.  S.,  in  the 
Biblical  Review,  1848.  III.  Review  of  my  Formal  Logic  (since  acknowledged  by 
Mr.  Mansel,  the  author  of  the  Prolegomena  Lpgica)  in  the  North  British  Review/or 
May  1851,  No.  29.  IV.  Discussions  on  Philosophy  by  Sir  W.  Hamilton;  London, 
Longman  and  Co.;  Edinburgh,  Machlachlan  and  Co.  (1st  edition,  1852,  8vo.  pp. 
621*-652*  ;  2nd  ed.  1853,  8vo.  pp.  676-707)  ;  in  which  a  letter  is  reprinted  which 
first  appeared  in  the  Athoneettm  of  August  24,  1850. 


The  relations  of  my  work  on  Formal  Logic  to  the  present 
syllabus  are  as  follows.  Chapter  I,  First  Notions,  may  afford  previous 
knowledge  to  the  student  who  has  hitherto  paid  no  attention  to  the 
subject.  Chapter  III,  On  the  abstract  form  of  the  Proposition  may 
be  consulted  at  §  93  of  this  syllabus.  Chapters  IV,  On  Propositions, 
and  V  and  VI,  On  the  Syllogism,  are  rendered  more  easy  by  the  nota- 
tion of  this  syllabus,  and  are  partially  superseded.  Chapter  XIV, 
On  the  verbal  Description  of  the  Syllogism,  is  entirely  superseded. 

rest  of  the  work  may  be  read  as  the  titles  of  the  chapters 

A  syllabus  deals  neither  in  development  nor  in  diversified  ex- 
ample :  and  does  not  make  the  space  occupied  by  any  detail  a 
measure  of  its  importance  as  a  part  of  the  whole.  I  have  omitted 
many  subjects  which  are  to  be  found  in  all  the  books,  or  dwelt  lightly 
upon  them  :  partly  because  more  detail  is  contained  in  my  Formal 
Logic,  partly  because  any  one  who  masters  this  tract  will  be  able  to 
judge  for  himself  what  I  should  have  written  on  the  omitted  subjects. 
I  have  also  endeavoured  to  remember  that  as  a  work  of  this  kind 
proceeds,  less  detail  of  explanation  is  necessary. 

I  should  suppose  that  a  student  of  ordinary  logic  would  find  no 
great  difficulty  in  understanding  my  meaning  :  and  that  those  who 
are  accustomed  to  symbolic  expression,  mathematical  or  not,  would, 
even  though  unused  to  logical  study,  find  no  more  difficulty  than  an 
ordinary  student  finds  in  Aldrich's  Compendium.  Either  of  these 
classes,  I  should  think,  would  not  fail  to  come  to  the  point  of  under- 
standing at  which  a  reflecting  mind  can  allow  itself  to  meditate 
acceptance  or  rejection  without  latent  fear  of  over-confidence. 
Whether  a  beginner  who  is  conversant  neither  with  ordinary  logic 
nor  with  symbolic  language  will  comprehend  me  is  another  question  : 
and  one  on  which  those  who  try  will  divide  into  more  than  two 
classes.  Such  a  reader,  making  concrete  examples  for  himself  as  he 
goes  on,  and  never  leaving  any  article  until  he  has  done  this,  will 
either  get  through  the  whole  tract,  or  will  stop  at  the  precise  point  at 
which  he  ought  to  stop,  upon  the  principle  of  the  next  paragraph. 

Every  spoon  has  some  mouths  that  it  can  feed  ;  and  some  that  it 
cannot.  Every  writer  has  some  readers  who  are  made  for  him,  and 
he  for  them  ;  and  some  between  whom  and  himself  there  is  a  great 
gulph.  I  might  prove  this,  in  my  own  case,  by  a  chain  of  discordant 


testimonies  running  through  thirty  years,  if  I  had  leisure  and  liking 
to  hunt  up  extracts  from  reviews.  I  will  content  myself  with  a 
couple  which  are  at  hand :  observing  that  I  have  no  acquaintance 
with  the  authors.  In  1830,  I  published  my  treatise  on  Arithmetic, 
and  the  following  sentences  speedily  appeared  in  reviews  of  it : — 

This  book  appears  to  us  to  mystify  a 
very  simple  science. 

It  is  as  clear  as  Cobbett  in  bis  lucid 

In  1847,  I  published  my  Formal  Logic,  and  two  opponents  of- my 
views  wrote  as  follows : 

Mr.  De  Morgan  I  This  is  an  undeniably  long  extract,  and  yet  we  would,  did  our 
is  certainly  not  a  j  limSts  allow,  continue  it  ....  "We  beg  the  reader's  notice  to 
lucid  writer.  the  exquisite  precision  of  its  language ;  to  the  definiteness  of 

every  line  in  the  picture  ;  for  though  it  is  a  description  of  a 
profound  mental  process,  still  it  is  a  luminous  picture,  the  light 
of  which  does  not  interfuse  its  lineaments  ....  We  are  at 
very  solemn  issue  with  Mr.  De  Morgan  upon  this  argument  .  .  . 

These  antagonisms  remind  us  of  the  stork  and  the  fox,  and  of  the 
failure  of  their  attempts  to  entertain  each  other  at  dinner.  When  an 
author's  dish  and  a  reader's  beak  do  not  match,  they  must  either 
divide  the  blame,  or  agree  to  throw  it  on  that  exquisite  piece  of 
atheism,  the  nature  of  things. 

The  points  on  which  I  differ  from  writers  on  logic  are  so  many 
and  so  fundamental,  that  I  am  among  them  as  Hobbes  among  the 
geometers,  and,  mutato  nomine,  may  say  with  him  of  Malmesbury  : — 
In  magno  quidem  periculo  versari  video  escistimationem  meam,  qui  a 
logicis  fere  omnibus  dissentio.  Eorum  enim  qui  de  iisdem  rebus 
mecum  aliquid  ediderunt,  aut  solus  insanio  ego  aut  solus  non  insanio  ; 
tertium  enim  non  est,  nisi  (quod  dicet  forte  aliquis)  insaniamus 

Hobbes  was  in  the  wrong :  the  question  of  parallel  or  contrast 
must  be  left  to  time.  But  though  some  writers  on  logic  explicitly 
renounce  me  and  all  my  works,  yet  one  point  of  those  same  works  is 
adopted  here  and  one  there ;  so  that  in  time  it  may  possibly  be  said 


Mahometans  eat  up  the  hog. 

All  these  differences  are  not  about  the  truth  or  falsehood  of  my 
neologisms,  but  about  the  legitimacy  of  their  adoption  into  logic,  the 
study  of  the  laws  of  thought.  And  I  cannot  imitate  Hobbes  so  far 


as  to  write  contra  fastum  logicorum,  seeing  that,  oblitis  obliviscendis, 
I  have  personally  nothing  but  courtesy  to  acknowledge  from  all  the 
writers  of  known  name  who  have  done  me  the  honour  of  alluding  to 
my  speculations.  In  return,  I  endeavour  to  tilt  at  their  shields  and 
not  at  their  faces.  Should  I  make  any  one  feel  that  I  have  missed 
my  attempt,  I  will  pray  the  introduction  into  these  lists  of  the  old 
practice  of  the  playground.  When  I  was  a  boy,  any  offence  against 
the  rules  of  the  game  was  held  to  be  nullified  if  the  offender,  before 
notice  taken,  could  cry  Slips! — as  I  now  do  over  all,  to  meet  con- 
tingencies. As  to  the  matter  of  our  differences,  I  neither  give  nor 
take  quarter :  it  is 

I  will  lay  on  for  Tusculum, 

Do  thou  lay  on  for  Rome  ! 

as  will  sufficiently  appear  in  my  notes. 

Now  to  another  topic.  I  produce  a  fragment  of  a  well-known 
conversation :  those  who  choose  may  fill  up  the  chasms. 

"...  I  therefore  dressed  up  three  paradoxes  with  some  ingenuity  ....  The 
whole  learned  world,  I  made  no  doubt,  would  rise  to  oppose  my  systems  :  but  then  I 
was  prepared  to  oppose  the  whole  learned  world.  Like  the  porcupine,  I  sat  self- 
collected,  with  a  quill  pointed  against  every  opposer.  Well  said,  my  boy,  cried  I, 
....  you  published  your  paradoxes;  well,  and  what  did  the  learned  world  say  to 
your  paradoxes  ?  Sir,  replied  my  son,  the  learned  world  said  nothing  to  my  para- 
doxes ;  nothing  at  all,  sir.  Every  man  of  them  was  employed  in  praising  his  friends 
and  himself,  or  condemning  his  enemies  ;  and  unfortunately,  as  I  had  neither,  I 
suffered  the  cruellest  mortification,  neglect." 

A  friend  of  the  author  just  quoted  remarked  that  a  shuttlecock 
cannot  be  kept  up  unless  it  be  struck  from  both  ends.  I  was  spared 
all  the  mortification  of  neglect  by  the  eminence  of  the  player  who 
took  up  the  other  battledore.  Of  this  celebrated  opponent  I  can  truly 
say  that,  so  far  as  I  myself  was  concerned,  I  never  looked  with  any- 
thing but  satisfaction  upon  certain  points  of  procedure  to  which  I 
shall  only  make  distant  allusion.  For  I  saw  from  the  beginning  that 
he  was  playing  my  game,  and  raising  the  wind  which  was  to  blow 
about  the  seeds  of  my  plant.  The  mathematicians  who  have  written 
on  logic  in  the  last  two  centuries  have  been  wholly  unknown  to  even 
the  far-searching  inquirers  of  the  Aristotelian  world :  to  the  late  Sir 
William  Hamilton  of  Edinburgh  I  owe  it  that  I  can  present  this  tract 
to  the  moderately  well  informed  elementary  student  of  logic,  as  con- 
taining matters  of  which  he  is  likely  enough  to  have  heard  something, 
and  may  possibly  be  curious  to  hear  more. 


In  controversy  —  and  controversy  was  to  him  an  element  of  life 
and  a  spring  of  action  —  Sir  William  Hamilton  was  too  much  the 
fencer  of  the  moment,  too  much  the  firer  of  to-morrow's  article  :  his 
impulses  sometimes  leap  him  over  the  barrier  which  divides  philosophy 
from  philosophism.  Hoot  the  proofs  of  this  out  of  his  pages,  and  we 
have  before  us  a  man  both  learned  and  ingenious,  profound  and  acute, 
weighty  and  flexible,  displaying  a  most  instructive  machinery  of 
thought  as  well  when  judged  right  as  when  judged  wrong,  save  only 
when  cause  of  regret  arises  that  Oxford  did  not  demand  of  his  youth 
two  books  of  Euclid  and  simple  equations.  In  describing  a  character 
some  points  of  which  are  at  tug  of  war  —  especially  when  liking  for 
quotation  was  one  of  them  —  Greek  may  meet  Greek.  He  showed 
more  fondness  than  was  politic  in  one  so  plainly  destined  to  survive 
the  grave  for  a  too  literal  version  of  the  motto 

But  on  the  other  hand,  though  too  much  inclined  to  rule  the  house, 

he    was    d/xoSss-jroTJi?    oW<?    Ix^esAAs*     be.    rov    $*iF»vgov    etvrov    Kattct,    x.»t 

Trx^ettei.  And  this  as  to  words  as  well  as  things.  Magnificent  com- 
mand of  language,  old  terms  rare  enough  to  be  new,  and  new  terms 
good  enough  to  be  old,  mask  defects  and  heighten  merits.  In  his 
writings  against  me,  it  delighted  him  to  enliven  the  statements  of  the 
accuser  by  portraits  of  the  mind  of  his  opponent  :  colouring  his  notion 
of  the  mathematician  from  the  darkness  of  his  want  of  notion  of  the 
mathematics,  his  great  and  admitted  defect  as  a  psychologist.  I  dealt 
with  the  statements  in  my  last  Cambridge  paper  :  I  now  oppose  my 
sketch  of  him  to  his  sketch  of  me,  without  the  least  misgiving  as  to 
which  of  the  two  will  be  pronounced  the  best  likeness. 


November  12,  1859. 


***  The  references  are  to  the  sections. 

Objective  View. —  (1-4)  Definitions,  &c.  (5-32)  The  arithmetical  form 
of  the  cumular  proposition.  (33-53)  The  arithmetical  form  of  the  cumular 
syllogism.  (54-56)  The  sorites.  (57-62)  Proposition  and  syllogism  of 
terminal  precision.  (63-73)  Exemplar  proposition  and  syllogism.  (74-86) 
Numerically  definite  proposition  and  syllogism.  (87-104)  Doctrine  of  figure 
and  wider  views  of  the  copula.  (105-111)  Aristotelian  syllogism. 

Subjective  View.—  (112-123)  Class  and  attribute.  (124-135)  Aggrega- 
tion and  composition.  (136-151)  Proposition,  judgment,  inference,  demon- 
stration. (152-173)  Relation:  onymatic  relations.  (174,  175)  The  four 
readings  of  a  proposition.  (176-189)  Mathematical  proposition  and  syllo- 
gism, (190-205)  Metaphysical  proposition  and  syllogism.  (206)  Arith- 
metical reading  in  intension.  (207-222)  Extent  and  intent.  (223-226) 
Hypothetical  syllogism,  &c.  (227-242)  Belief,  probability,  testimony,  argu- 
ment. (243)  Aggregation  and  composition  of  probabilities.  (243)  Symbols. 

Controversial  Notes. —  (1)  Use  of  logic.  (14)  Some  may  be  all,  denied  in 
practice  by  some  logicians.  (23)  Use  of  analysis  of  enunciation.  (27)  Con- 
version. (64)  Sir  W.  Hamilton's  system  ;  his  account  of  the  author's 
system.  (69)  Limitation  of  matter  of  a  syllabus.  (71)  Sir  W.  Hamilton's 
system.  (72)  Example  of  error  in  use  of  exemplar  system.  (93)  Logician's 
mode  of  supplying  the  defects  of  his  syllogism.  (96)  Incompleteness  of 
common  logic :  legitimate  subtleties.  (104)  Use  of  generalisation.  (108) 
References  for  old  logic.  (116)  The  world  has  got  beyond  the  logicians. 
(124)  Logical  and  physical  composition.  (131)  The  logician's  areal  phraseo- 
logy. (138)  The  logician's  form  and  matter.  (139)  Contrary  and  contra- 
dictory. (146)  The  logicians  and  the  mathematicians.  (163)  Can  the 
principles  of  conversion  and  transition  be  deduced  from  those  of  identity, 
difference,  and  excluded  middle?  (165)  The  logician's  form  and  matter. 
(170)  For  logician's  aggregation  read  composition.  (172)  Metaphysical 
notions,  why  introduced  in  logic.  (175)  Dependence  of  a  proposition  upon 
its  universe.  (177)  Genus  and  species.  (190)  Metaphysical  terms.  (202)  _ 
Predicables.  (210)  Wrong  opposition  in  universal  and  particular.  (211) 
Extent  and  intent.  (212)  Scope  and  force.  (214)  The  modern  logician's 
extension  and  comprehension.  (216)  A  word  supplied  by  Sir  W.  Hamilton. 
(220)  Class  and  attribute.  (228)  Belief.  (232)  Bias  and  exhortations  to 
get  rid  of  it  considered. 

***  The  notes  end  the  articles  in  which  the  marks  of  reference  occur. 


1.  LOGIC  *  analyses  the  forms,  or  laws  of  action,  of  thought. 

*  Logic  has  a  tendency  to  correct,  first,  inaccuracy  of  thought,  secondly, 
inaccuracy  of  expression.  Many  persons  who  think  logically  express  them- 
selves illogically,  and  in  so  doing  produce  the  same  effect  upon  their  hearers 
or  readers  as  if  they  had  thought  wrongly.  This  applies  especially  to  teachers, 
many  of  whom,  accurate  enough  in  their  own  thoughts,  nourish  sophism  in 
their  pupils  by  illogical  expression. 

It  is  very  commonly  said  that  studies  which  exercise  the  thinking  faculty, 
and  especially  mathematics,  are  means  of  cultivating  logic,  and  may  stand 
in  place  of  systematic  study  of  that  science.  This  is  true  so  far,  that  every 
discipline  strengthens  the  logical  power :  that  is  to  say,  strengthens  most  of 
what  it  finds,  be  the  same  good  or  bad.  It  is  further  true  that  every  disci- 
pline corrects  some  bad  habits :  but  it  is  equally  true  that  every  discipline 
tends  to  confirm  some  bad  habits.  Accordingly,  though  every  exercise  of 
mind  does  much  more  good  than  harm,  yet  no  person  can  be  sure  of  avoiding 
the  harm  and  retaining  only  the  good,  except  by  that  careful  examination  of 
his  own  mental  habits  which  most  often  takes  place  in  a  proper  study  of  logic, 
and  is  seldom  made  without  it.  This  being  done,  and  the  house  being  built, 
the  scaffolding  may  be  thrown  away  if  the  builder  please :  though  in  most 
cases  it  will  be  advisable  to  keep  it  at  hand,  for  use  when  inspection  or 
repairs  are  needed.  Some  persons  make  the  argument  about  the  utility  of 
logic  turn  on  .the  question  whether  disputation  is  or  is  not  best  conducted 
syllogistically :  on  which  I  hold — waiving  the  utter  irrelevancy  of  the 
question — that  those  who  cannot  so  argue  need  to  learn,  while  those  who  can 
have  no  need  to  practise.  It  is  just  the  same  with  spelling. 

As  to  the  difficult  word  form,  and  the  variations  of  it,  I  refer  to  Dr. 
Thomson's  Outlines  of  the  Necessary  Laws  of  Thought,  §  11-15.  By  a  form 
of  thought  I  mean  a  necessary  law  of  action,  considered  independently 
of  any  particular  matter  of  thought  to  which  it  is  applied. 

2.  Logic  is  formal,  not  material:-  it  considers  the  law  of 
action,  apart  from  the  matter  acted  on.  It  is  not  psychological, 
not  metaphysical :  it  considers  neither  the  mind  in  itself,  nor  the 
nature  of  things  in  itself;  but  the  mind  in  relation  to  things,  and 
things  in  relation  to  the  mind.  Nevertheless,  it  is  so  far  psycho- 


10  SUBDIVISIONS. — NAMES.  [2-6. 

logical  as  it  is  concerned  with  the  results  of  the  constitution  of  the 
mind :  and  so  far  metaphysical  as  it  is  concerned  with  the  right 
use  of  notions  about  the  nature  and  dependence  of  things  which, 
be  they  true  or  be  they  false,  as  representations  of  real  existence, 
enter  into  the  common  modes  of  thinking  of  all  men. 

3.  The  study  of  elementary  logic  includes  the  especial  con- 
sideration of — 

1.  The  term  or  name,  the  written  or  spoken  sign  of  an 

object  of  thought,  or  of  a  mode  of  thinking. 

2.  The  copula  or  relation,   the   connexion   under  which 

terms  are  thought  of  together. 

3.  The  proposition,  terms  in  relation  with  one  another ; 

and  the  judgment,  the  decision  of  the  mind  upon  a 
proposition:  usually  joined  in  one,  under  one  or 
other  of  the  names. 

4.  The  syllogism,  deduction  of  relation  by  combination  of 

other  relations. 

4.  The  thing  which  is  not  of  the  mind,  and  can  be  imagined 
to  exist  without  the  mind,  is  the  object :  the  mind  itself  is  called 
the  subject  of  that  object.     Thus  even  a  relation  between  two 
minds  may  be  an  object  to  a  third  mind.     Logic  considers  only 
the  connexion  of  the  subjective  .and  objective:  it  treats  of  things 
non  secundum  se,  sed  secundum  esse  quod  habent  in  anima. 

5.  The  consideration  of  names,  as  names,  may  be  made  to  fur- 
nish the  key  to  the  mechanical,  or  instrumental,  treatment  of  the 
ordinary  proposition  and  the  ordinary  syllogism. 

6.  For  this  purpose  a  name  is  a  mere  mark,  attached  to  an 
object:  a  letter  painted  on  a  post  would  do  as  well  for  expla- 
nation as  the  name  of  a  notion  or  concept  in  the  mind  attached  in 
thought  to  an  external  object.     In  this  part  of  the.  subject,  the 
assertion  *  Every  man  ig  an  animal '  is  treated  as  if  it  were  merely 
'  Every  object  which  has   the  name  m-a-n  has  also  the  name 
a-n-i-m-a-l.'     It  answers*  to  *  Every  post  on  which  X  is  painted 
has  also  Y  painted  on  it.' 

*  Remember  that  in  producing  a  name,  the  existence  of  things  to  which 
it  applies  is  predicated,  i.  e.  asserted  :  and  (§  16)  existence  in  the  universe 
of  the  propositions.  There  is  no  conditional  proposition  intended,  as  Every 
X  is  Y  (if  Y  exist).  Every  proposition  is  a  conceivable  or  imaginable  truth, 
when  its  terms  are  conceivable  or  imaginable,  except  only  when  it  announces 
a  contradiction,  as '  Some  men  are  not  men,'  or  when  it  is  its  own  subject- 
matter  and  denies  itself,  as  '  What  I  now  say  is  false,'  a  proposition  which  is 
false  if  it  be  true,  and  true  if  it  be  false. 


7.  The  proposition,  in  this  view,  is  no  more  than  the  con- 
nexion of  name  with  name,  as  marks  of  the  same  object:  the 
judgment  is  no  more  than  assertion  or  denial  of  that  connexion. 

The  word  is  asserts  the  connexion :  the  words  is  not  deny  it. 

8.  This  kind  of  proposition  belongs  to  the  arithmetical  view  of 
logic :  there  is  in  it  result  of  enumeration  of  similar  instances :  as 
in  '  Every  X  is  Y ', '  50  Xs  are  not  Ys ',  i.e.  there  are  50  (or  more) 
instances  in  which  the  name  X  occurs  unassociated  with   the 
name  Y. 

9.  The  first-mentioned  name  is  called  the  subject :  the  second 
the  predicate. 

10.  The  logical  quantity  (i.e.  number  of  instances)  is  either 
definite,  or  more  or  less  vague. 

11.  Definite  quantity  is  either  absolutely  or  relatively  definite: 
absolutely,  as  in  *  50  Xs  (and  no  more)  are  Ys ' ;  relatively,  as  in 
'  2-sevenths  (and  no  more)  of  all  the  Xs  are  Ys ',  and  in  '  All  the 
Xs  are  Ys ',  and  in  *  None  of  the  Xs  are  Ys ',  which  last  is  both 
absolutely  and  relatively  definite. 

12.  Quantity  may  be  definite  at  one  end,  and  vague  at  the 
other :  as  in  ( 50  (or  more)  Xs  are  Ys ' ;  'at  least  2-sevenths  of 
all  the  Xs  are  Ys ' ;  *  most  of  the  Xs  (i.  e.  more  than  half)  are 
Ys ',  which,  however,  is  limited  at  one  end,  not  dejinite. 

13.  The  only  perfectly  definite  quantities  in  ordinary  use  are 
all  and  none.     The  materials  for  other  absolute  or  relative  nume- 
rical definiteness  are  but   seldom  found  in   human   knowledge. 
The  words  all  and  none  are  signs  of  total  quantity,  and  make 
propositions  universal,  as  '  All  Xs  are  Ys  ',  '  No  Xs  are  Ys '. 

14.  The  contrary  (usually  called  contradictory}  propositions  of 
the  last  are  '  Some  Xs  are  not  Ys ',  and  *  Some  Xs  are  Ys '. 
Here  f  some '  is  a  quantity  entirely  vague  in  one  direction :  it  is 
not-none ;  one  *  at  least ;    or  more ;  all,f  it  may  be.      Some,  in 
common  life,  often  means  both  not-none  and  not-all;  in  logic,  only 
not-none.     Some  is  the  mark  of  partial  quantity ;  and  the  propo- 
sitions which   commence  with  it   are  particular.     The  contrary 
(usually  contradictory)  forces  of  the  pairs  are  seen  in  '  Either  all 
Xs  are  Ys,  or  some  Xs  are  not  Ys ;  not  both ' ;  and  in  *  Either  no 
Xs  are  Ys,  or  some  Xs  are  Ys ;  not  both.' 

*  Kemember  that  some  does  not  guarantee  more  than  one.  There  is 
much  distinction  between  none,  (no  one)  as  the  logical  contrary  of  some 
(one  or  more),  and  nothing  as  the  limit  of  some  quantity.  In  passing  from 
the  proposition  '  no  man  can  live  without  air'  to  'there  is  one  man  at  least 
who  can  live  without  air'  we  make  a  transition  which  alters  the  notions 

12  QUANTITY. — UNIVERSE.  [14-16. 

or  concepts  attached  to  man :  the  word  man  no  longer  represents  entirely 
the  same  idea.  But  in  passing  from  an  indenture  of  apprenticeship  with  no 
premium  at  all  to  one  with  a  premium  of  one  farthing,  we  make  no  change 
of  notion.  An  Act  was  once  passed  exempting  such  indentures  from  duty 
when  the  premium  was  under  five  pounds  sterling :  the  Court  of  King's 
Bench  held  that  the  exemption  did  not  apply  when  there  was  no  premium  at 
all,  because  "  no  premium  at  all "  is  not  "  a  premium  under  five  pounds." 
That  is,  the  judges  gave  to  nothing,  as  the  terminus  of  continuous  quantity, 
the  force  of  none,  as  the  logical  contrary  of  some;  and  violated  to  the 
utmost  the  principle  of  the  Act,  which  was  intended  as  a  relief  to  those 
who  could  only  pay  small  premiums,  and  a  fortiori,  or  rather  a  fortissimo, 
to  those  who  could  pay  none  at  all.  Lawyers  ought  to  be  much  of  logicians, 
and  something  of  mathematicians. 

f  Some  may  be  all.  In  common  language  this  is  or  is  not  the  case 
according  to  the  speaker's  state  of  knowledge.  But  in  logic  there  are  no 
implications  which  depend  upon  the  matter.  When  a  logician  says  that 
'Every  X  is  Y'  he  means  that  '  All  the  Xs  are  some  Ys'  and  that  'some 
Ys  are  all  the  Xs'.  Whether  he  have  or  have  not  exhausted  all  the  Ys 
he  does  not  here  profess  to  state,  even  if  he  know.  Again,  when  he  says 
'  Some  Xs  are  Ys'  he  does  not  mean  to  imply  '  Some  Xs  are  not  Ys':  that 
is,  his  some  may  be  all,  for  anything  he  asserts  to  the  contrary.  But  when 
two  propositions,  each  of  which  contains  the  vague  some,  are  conjoined,  the 
mere  meaning  may  render  the  conjunction  an  absurdity  unless  some  take  the 
force  of  all.  Just  as  in  algebra  an  equation  having  two  unknown  quantities 
has  the  values  of  those  quantities  vague ;  but  when  two  such  equations  are 
conjoined,  those  values  become  definite  :  so  in  logic,  in  which  the  same  thing 
occurs.  Thus  the  two  propositions  'All  Xs  are  some  Ys',  and  'All  Ys  are 
some  Xs',  when  true  together,  force  the  inference  that  some  must  in  both 
cases  be  all.  Forget  that  some  is  that  which  may  be  all,  and  these  two 
propositions  appear  to  contradict  one  another :  very  distinguished  logicians 
have  asserted  that  they  do  so.  Sir  W.  Hamilton  (Discussions,  2nd  edition, 
p.  688)  calls  them  incompossible  propositions,  meaning  propositions  which 
cannot  be  true  together.  The  word  is  an  excellent  one,  and  much  wanted : 
but  not  here.  These  two  propositions  are  not  incompossible,  unless  some 
take  into  its  meaning  not-all  as  well  as  not-none  :  and  some  is  never  allowed 
by  logicians  to  mean  not-all.  It  is  needless  to  argue  things  so  plain :  it  would 
have  been  needless  to  state  them  twice,  except  for  the  eminence  of  the 
writers  who  deny  them  upon  occasion. 

15.  In  f  All  Xs  are  Ys',  Y  is  partially  spoken  of;  there  may 
or  may  not  be  more  Ys  besides :  the  same  of '  Some  Xs  are  Ys '. 
In  '  No  Xs  are  Ys ',  and  in  '  Some  Xs  are  not  Ys ',  Y  is  totally 
spoken  of;  each  X  spoken  of  is  not  any  one  of  all  the  Ys. 

16.  By  the  universe  (of  a  proposition)  is  meant  the  collection 
of  all  objects  which  are  contemplated  as   objects  about  which 
assertion  or  denial  may  take  place.     Let  every  name  which  belongs 
to  the  whole  universe  be  excluded  as  needless :  this  must  be  parti- 
cularly remembered.     Let  every  object  which  has  not  the  name 


X  (of  icldch  there  are  always  some)  be  conceived  as  therefore 
marked  with  the  name  x,  meaning  not-X,  and  called  the  contrary 
of  X.  Thus  every  thing  is  either  X  or  x  ;  nothing  is  both :  '  All 
Xs  are  Ys '  means  '  No  Xs  are  ys ' :  '  No  Xs  are  Ys '  means  (  All 
Xs  are  ys ' :  '  Some  Xs  are  not  Ys '  means  •'  Some  Xs  are  ys ' : 

*  Some  Xs  are  Ys '  means  '  Some  Xs  are  not  ys '. 

17.  The  following  enlarged  definitions  include  the  definitions 
above  given,  and  apply  to  all  the  uses  of  terms  and  relations  in 
this  work.     Let  a  TERM  be  total  or  partial  according  as  every 
existing  instance  must  or  need  not  be  examined  to  verify  the 
proposition.     Thus  in  e  Everything  is  either  X  or  Y ',  X  and  Y 
are  both  partial :  an  object  being  examined  and  found  to  be  X, 
the  proposition  is  made  good  so  far  as  that  object  is  concerned ; 
that  object  may  also  be  Y,  but  if  so,  it  need  not  be  ascertained : 
consequently,  Y  is  partially  spoken  of;  and  the  same  may  be  said 
of  X.     But  in  '  Some  things  are  neither  Xs  nor  Ys ',  X  and  Y 
are  both  total :  we  can  only  verify  it  by  an  object  which  is  not 
any  one  of  the  Xs  and  not  any  one  of  the  Ys. 

18.  Let  a  PROPOSITION  be  universal  or  particular,  according  as 
the  whole  universe  of  objects  must  or  need  not  be  examined  to 
verify  it.     Thus  f  Everything  is  either  X  or  Y '  is  plainly  uni- 
versal :  but f  Some  things  are  neither  Xs  nor  Ys '  is  particular : 
the  first  object  examined  may  settle  the   truth   of  the   propo- 

19.  Let  a  PROPOSITION  be  affirmative  which  is  true  of  X  and  X, 
false  of  X  and  not-X  or  x ;  negative,  which  is  true  of  X  and  x, 
false  of  X  and  X.     Thus  ( Every  X  is  Y '  is  affirmative :  *  Every 
X  is  X '  is  true ;  ( Every  X  is  x '  is  false.     But  f  Some  things  are 
neither  Xs  nor  Ys  '  is  also  affirmative,  though  in  the  form*  of  a 
denial :  ' Some  things  are  neither  Xs  nor  Xs '  is  true,  though 
superfluous  in  expression  ;  '  Some  things  are  neither  Xs  nor  xs '  is 
false.     Again,  *  Everything  is  either  X  or  Y '  is  negative,  though 
in  the  form  of  an  assertion :  ( Everything  is  either  X  or  X '  is 
false  ;  '  Everything  is  either  X  or  x  '  is  true. 

*  When  contrary  terms  are  introduced,  it  is  impossible  to  define  the 
opposition  of  quality  by  assertion  or  denial :  for  every  assertion  is  a  denial, 
and  every  denial  is  an  assertion.  The  denial  'No  X  is  Y'  is  the  assertion 

*  All  Xs  are  ys.'     The  necessary  distinction  between  affirmative  and  negative 
is  therefore  drawn  as  in  the  text :  these  technical  terms  are  retained,  though 
perhaps  they  are  hardly  the  right  ones  for  me  to  use. 

20.  Affirmative  and  negative  propositions  are  said  to  be  of 
different  quality. 

14  FORMS  OF  ENUNCIATION.  [21-23. 

21.  Let  X,  totally  spoken  of,  be  X)  or  (X  :  let  X,  partially 
spoken  of,  be  )X  or  X(.  Let  a  negative  proposition  be  denoted 
by  one  dot;  an  affirmative  proposition  by  two  dots  or  none,  at 
pleasure.  I  follow  Sir  William  Hamilton  in  calling  this  notation 
spicular  (see  §  216,  note).  So  far  as  yet  appears,  we  have  pro- 
positions with  the  following  symbols, — 

Universal  Affirmative  X))Y      All  Xs  are  some  Ys. 
Particular  Negative       X(-(Y      Some  Xs  are  not  (all)  Ys. 

Universal  Negative        X)-(Y      All  Xs  are  not  (all)  Ys. 
Particular  Affirmative  X(  ) Y      Some  Xs  are  some  Ys. 




22.  These  forms  have  been  denoted  by  the  letters  A,  O,  E,  I, 
for  many  centuries ;  A  and  I  from  the  vowels  in  hffvrmo ;  E  and 
O  from  the  vowels  in  n&gQ.     The  word  (all),  in  parentheses,  is 
not  grammatical:    the  word  any  should  be  substituted  for  all. 
The  reason  why,  for  the  present,  I  do  not  use  f  any '  will  appear  in 
the  sequel. 

23.  Take  the  four  pairs,  X,  Y;  X,y ;  x,  Y;  x,  y;  and  apply 
the  four  forms  above  to  all  four.      Sixteen  results  appear;    of 
which  eight  are  but  a  repetition  of  the  other  eight.    Of  the  eight* 
which  are  distinct,  we  have  four  written  above :  the  remaining 
four  appear  among  the  following, — 

From  ))  we  have  X))Y,  X))y,  x))Y,  and  x))y.  Of  these 
X))y  is  obviously  X)-(Y.  And  x))Y  is  x)-(y,  a  new  form :  no 
not-X  is  not-Y ;  nothing  is  both  not-X  and  not-Y  ;  everything  is 
either  X  or  Y.  This  being  a  universal  proposition  with  both 
terms  partial,  and  also  a  negative  proposition,  let  it  be  marked 
X(-)Y.  Again,  x))y  is  x)'(Y,  or  Y)-(x,  or  no  Y  is  not-X,  or 
every  Y  is  X,  or  some  Xs  are  all  Ys,  or  X((Y. 

From  ()  we  have  XQY,  XQy,  x()Y,  x()y.  Here  X(  )y 
is  X(-( Y ;  and  x(  )Y  is  Y(  )x,  some  Ys  are  not  Xs,  or  all  Xs  are 
not  some  Ys,  or  X)-)Y.  And  x()y  is  a  new  form,  Some  not-Xs 
are  not-Ys ;  some  things  are  neither  Xs  nor  Ys.  This  is  a  par- 
ticular proposition,  affirmative,  with  X  and  Y  both  total :  let  it  be 
marked  X)(Y. 

*  Any  one  who  wishes  to  test  himself  and  his  friends  upon  the  question 
whether  analysis  of  the  forms  of  enunciation  would  be  useful  or  not,  may  try 
himself  and  them  on  the  following  question: — Is  either  of  the  following 
propositions  true,  and  if  either,  which  ?  1.  All  Englishmen  who  do  not  take 
snuff  are  to  be  found  among  Europeans  who  do  not  use  tobacco.  2.  All 
Englishmen  who  do  not  use  tobacco  are  to  be  found  among  Europeans  who 
do  not  take  snuff,  llequired  immediate  answer  and  demonstration. 




24.  The  eight  distinct*  forms  in  which  X  and  Y  appear 
are  as  follows;  the  ungrammatical  introduction  (all)  being  made 
as  before, — 

Universal  propositions  Contrary  particular  propositions 

X))Y    All  Xs  are  some  Ys  X(-(Y    Some  Xs  are  not  (all)  Ys 

X)-(Y   All  Xs  are  not  (all)  Ys  XQY 

X(-)Y   Everything  is  either  some  X      X)(Y 

or  some  Y  (or  both) 
X((Y     Some  Xs  are  all  Ys  X)-)Y 

For  symmetry,  X)-(Y  might  be  read  '  Everything  is  either  not 
(all)  X,  or  not  (all)  Y ' ;  and  X(  )Y  as  *  Some  things  are  both 
some  Xs  and  some  Ys'.  This  will  be  better  seen  when  we 
come  to  §  206.  At  present,  however,  I  preserve  the  ordinary 

*  The  following  is  the  comparison  of  the  notation  in  my  formal  Logic 
with  that  used  in  my  second  and  third  papers  in  the  Cambridge  Transactions, 
and  in  this  syllabus. 

Some  Xs  are  some  Ys 
Some  things  are  not  either 

(all)  Xs  nor  (all)  Ys 
All  Xs  are  not  some  Ys 

For  X))Y, 
For  X((Y, 
For  X)-(Y, 
For  X(-)Y, 

X)Y  and  A, 
X(Y  and  A' 
X.Y  andE, 
x.y     and  E' 

For  X()Y, 
For  X)(Y, 
For  X(-(Y, 
For  X)-)Y, 

XY  and  I, 
xy     and  I' 
X:Y  and  O, 
Y:X  and  O' 

For  X»Y,  D,  and  A,  +  a 
For  X  ||  Y,  D  and  A/  +  A' 
For  X(c(Y,  D'  and  A'  +  O, 

For  X)o(Y,  C,  and  E,  +  I' 
For  X  |-|  Y,  C  and  E,  +  E' 
For  X(o)Y,  C'  and  E'  +  I, 

These  comparative  notations  being  fixed  in  the  mind,  any  part  of  my  Formal 
Logic  may  be  read  in  illustration  of  the  present  work.  And  the  detailed 
character  (§  64,  note)  of  the  latest  notation  is,  if  I  may  judge,  so  much  of  a 
facilitation,  that  any  reader  of  the  Formal  Logic  will  find  it  easier  to  trans- 
late the  notation  as  he  goes  on  than  to  confine  himself  entirely  to  the 
notation  of  that  work :  and  this  especially  as  to  the  tests  of  validity  and  the 
assignment  of  the  symbol  of  inference. 

25.  The  thirty-two  forms  which  arise  from   application   of 
contraries  are  as  now  written,  all  the  eight  cases  above  being 
used :  the  four  in  each  line  are  of  identical  meaning. 


X))Y  ,  X)-(y  ,  x(-)Y 
X)-(Y  X))y  x((Y 
X(-)Y  X((y  x))Y 
X((Y  X(-)y  x)-(Y 

26.  The  rule  of  contraversion — changing   a   name   into   its 
contrary  without  altering  the  import  of  the  proposition — is,  Change 
also  the  quantity  of  the  term,  and  the  quality  of  the  proposition. 
Thus  X))Y  is  X)-(y  and  x(-)Y.     When  both  names  are  contra- 






.  *»y 



X(-(y  , 


,  *)(y 



X)-)y  , 


,  *()y 



,    X)(y 


.  xC(y 




verted,  change  both  quantities,  and  preserve  the  quality :    thus 

27.  The  rule  of   conversion* — making   the   names  change 
places,  without  altering  the  import  of  the  proposition — is,  Write 
or  read  the  proposition  backwards.      Thus  X))Y  is  Y((X;    or 
X))Y  may  be  read  backwards,   Some  Ys  are  all  Xs.     That  is, 
make  both  the  terms  and  their  quantities  change  places. 

*  Writers  on  logic  have  nearly  always  meant  by  conversion  merely  the 
change  of  place  in  the  terms,  without  change  of  place  in  the  quantities.  Ac- 
cordingly, when  the  quantities  are  different,  (common)  logical  conversion 
is  illegitimate.  Thus  X))Y  and  Y))X  are  not  the  same :  but  X()Y  and 
YQX  are  the  same.  There  is  this  difficulty  in  the  way  of  using  the  word 
conversion  in  the  sense  proposed  in  the  text :  namely,  that  common  logic  has 
rooted  it  in  common  language  that  'Every  X  is  Y'  is  the  converse  (true 
or  false  as  the  case  may  be)  of  'Every  Y  is  X.'  Leaving  the  common 
idioms  for  the  student  to  do  as  he  likes  with,  I  shall,  if  I  have  occasion  to 
speak  of  a  proposition  in  which  terms  only  are  converted,  and  not  quantities, 
call  it  a  term- converse. 

28.  Each  universal  is  inconsistent  with  the  universals  of  dif- 
ferent qualities,  and  indifferent  to  the  universal  of  different  quan- 
tities.    Thus  X))Y  is  inconsistent  with  X)g(Y  and  X(-)Y,  and 
neither   affirms  nor   denies  X((Y.     Each  universal  affirms   the 
particulars  of  the   same   quality,  contradicts  the  particular   of 
different  quantities,  and  is  indifferent  to  the  particular  of  the  same 
quantities.      Thus  X))Y  affirms  X()Y   and  X)(Y,   contradicts 
X(-(Y,  and  neither  affirms  nor  denies  X)-)Y. 

Is  inconsistent        Neither  affirms 
with  nor  denies 

)•(      0 

Is  neither  affirmed  nor  denied  by 
(0       )(  ('(        )') 

)•(      0  )')      C( 

29.  Contrary  names,  in  identical  propositions,  always  appear 
with  different  quantities.     We  cannot  speak  of  some  Xs  without 
speaking  about  all  xs ;  nor  of  all  Xs  without  speaking  about 
some  xs. 

30.  A  particular  proposition  is  strengthened  into  a  universal 
which  affirms  it  (and  more,  may  be)  by  altering  one  of  the  quan- 
tities :  thus  )•)  is  affirmed  in  (•)  and  in  )•(     Remember  §  16. 



()        )( 


)')        C( 


)(        0 


Is  affirmed  by 


)'(       C) 


((        )) 


))        (( 


C)        )'( 


31-36.]  SYLLOGISM.  17 

31.  In  a  universal  proposition,  if  one  term  be  partial,  it  has 
the  amount,  not  the  character,  of  the  quantity  of  the  other:  if  both, 
the  quantities  of  the  two  terms  together  make  up  the  whole  uni- 
verse, with  the  part  common  to  both,  if  any,  repeated  twice. 

32.  In  a  particular  proposition,  the  quantity  of  a  partial  term 
is  vague,  but  remains  the  same  through  all  forms.     And  when 
both  terms  are  total,  the  partial  quantity  still  remains  expressed : 
as  in  X)(Y,  or  Some  things  are  neither  Xs  nor  Ys ;  which  some 
things  are  as   many  as  the  xs  or  ys  in  the  equivalents  x()y, 
X)-)y,andx(-(Y.    ' 

33.  If  a  proposition  containing  X  and  Y  be  joined  with  a 
proposition  containing  Y  and  Z,  a  third  proposition  containing  X 
and  Z  may  necessarily  follow.     In  this  case  the  two  first  pro- 
positions (premises)  and  the  proposition  which  follows  from  them 
(conclusion)  form  a  syllogism. 

34.  If  an  X  be  a  Y,  if  that  same  Y  be  a  Z,  then  the  X  is  the 
Z.     This  is  the  unit-syllogism  from  collections  of  which  all  the 
syllogisms  of  this  mode  of  treating  propositions  must  be  formed. 
At  first  sight  it  seems  as  if  there  were  another :  if  an  X  be  a  Y, 
if  that  same  Y  be  not  any  Z,  then  the  X  is  not  any  Z.     But  this 
comes  under  the  first,  as  follows :  the  X  is  a  Y,  that  Y  is  a  z, 
therefore  the  X  is  a  z,  that  is,  is  not  any  Z.     The  introduction  of 
contraries  brings  all  denials  under  assertions. 

35.  Two  premises  have  a  valid  conclusion  when,  and  only 
when,  they  necessarily   contain   unit-syllogisms;    and   the  con- 
clusion has  one  item   of  quantity  for   every  unit-syllogism  so 
necessarily  contained. 

36.  And  all  syllogisms  may  be  derived  from  the  following 
combinations : — 

))  ))  or  X))Y  Y))Z,  or  All  Xs  are  Ys  and  all  Ys  are  Zs. 
The  conclusion  is  X))Z,  All  Xs  are  Zs :  there  is  the  unit-syllogism, 
This  X  is  a  Y,  that  same  Y  is  a  Z,  repeated  as  often  as  there  are 
Xs  in  existence  in  the  universe.  Or,  X))Y))Z  gives  X))Z,  or 
))  ))  gives  )). 

()  ))  or  X()Y  Y))Z,  or  Some  Xs  are  Ys,  all  Ys  are  Zs. 
The  conclusion  is  X(  )Z,  Some  Xs  are  Zs :  there  is  the  unit- 
syllogism  so  often  as  there  are  Xs  in  the  first  premise.  Or, 
X()Y))Z  gives  X()Z,  or  ()  ))  gives  (). 

((  ()  or  X((Y  Y()Z,  or  Some  Xs  are  all  Ys,  some  Ys  are  Zs. 
The  conclusion  is  X(  )Z,  Some  Xs  are  Zs :  this  case  is,  as  to  form, 



nothing  but  the  last  form  inverted.     Or,  X((Y()Z  gives  X()Z  or 

((  ))  or  X((Y  Y))Z  or  Some  Xs  are  all  Ys,  All  Ys  are  Zs. 
The  conclusion  is  X()Z,  Some  Xs  are  Zs,  as  many  as  there  are 
Ys  in  the  universe.  Or,  X((Y))Z  gives  XQZ,  or  ((  ))  gives  (  ). 
But  this  case  gives  no  stronger  conclusion  than  ()  ))  or  than  ((  (), 
though  it  has  both  premises  universal. 

These  are  all  the  ways  in  which  affirmative  premises  produce 
a  conclusion  in  a  manner  which  has  no  need  to  take  cognisance  of 
the  existence  of  contrary  terms.  And  since  all  negations  are  con- 
tained among  affirmations  about  contraries,  we  may  expect  that 
application  of  these  cases  to  all  combinations  of  direct  and  contrary 
will  produce  all  possible  valid  syllogisms. 

37.  Apply  the  form  ))  ))  to  the  eight  varieties  XYZ,  xYZ, 
xyZ,  xyz,  xYz,  XYz,  Xyz,  XyZ,  and  contravert  x,  y,  z,  when- 
ever they  appear.     Thus  ))  ))  applied  to  x  y  Z  is  x))y))Z,  or 
X((Y  combined  with  Y(-)Z  or  X((Y(-)Z  or  ((  (•).      The  con- 
clusion is  x))Z  or  X(-)Z.     That  is,  X((Y(-)Z  gives  X(-)Z;   or, 
If  some  Xs  be  all  [make  up  all  the]  Ys,  and  everything  be  either 
Y  or  Z,  then  everything  is  either  X  or  Z.     This  process  applied 
to  the  eight  varieties  gives  the  following  eight  forms  of  universal 
syllogism,  that  is,  universal  premises  with  universal  conclusion. 

.  ))  ))   (0  »   ((  (•)   ((  ((    (•)  X  .  ))  X    X  ((    X  (•) 

Here  are  all  the  ways  in  which  two  universals  give  different  quan- 
tities to  the  middle  term. 

38.  Apply  ()  ))  to  the  eight  varieties  and  we  have  eight 
minor-particular  syllogisms,  particular  conclusion  with  the  minor 
(or  first)  premise  particular, 

()))   )•)))   )((•).)(((   )•)>(   OX   (•(((   ('((•) 

Here  are  all  the  ways  in  which  a  particular  followed  by  a  uni- 
versal give  different  quantities  to  the  middle  term. 

39.  Apply  ((  (),  and  we  have  eight  major-particular  syllo- 
gisms, particular  conclusion  with  the  major  (or  second)  premise 

((()   XO   )))•).  )))(   )•((•(.  (((•(   (•))(   (•)» 

Here  are  all  the  ways  in  which  a  universal  followed  by  a  par- 
ticular gives  different  quantities  to  the  middle  term. 

40.  Apply  ((  ))  and  we   have  eight   strengthened  particular 
syllogisms,  universal  premises  with  particular  conclusion. 

(())   >())   ))O   ))((   XX   ((>(   (•)((   (•)(•) 


Here  are  all  the  ways  in  which  two  universals  give  the  same 
quantity  to  the  middle  term. 

41.  There  are   64   possible  combinations,  of  which   the   32 
enumerated  give  inference.     The  remaining  32  may  be  found  by 
applying  the  eight  varieties  to  (  )  (( ,  ))  (  ) ,  (  )  )(  and  (  )  )  •) :  and 
in  no  case  does  any  inference  follow.     Thus  X()Y  and  Y((Z 
are  consistent  with  any  of  the  eight  relations  between  X  and  Z, 
which  should  be  ascertained  by  trial. 

42.  The  test  of  validity  and  the  rule  of  inference   are   as 
follows, — 

There  is  inference  1.  When  both  the  premises  are  universal ; 
2.  When,  one  premise  only  being  particular,  the  middle  term  has 
different  quantities  in  the  two  premises. 

The  conclusion  is  found  by  erasing  the  middle  term  and  its 
quantities.  Thus  )•(  (•)  gives  )••)  or  ))  (§  21).  That  is  ' No  X 
is  Y,  and  Everything  is  either  Y  or  Z '  gives  '  Every  X  is  Z '. 

43.  Premises  of  like  quality  give  an  affirmative  conclusion :  of 
different  quality,  a  negative.     A  universal  conclusion   can   only 
follow  from  universals  with  the  middle  term  differently  quan- 
tified in  the  two.     From  two  particular  premises  nothing  can 

44.  A  particular  premise  having  the  concluding  term  strength- 
ened, the  conclusion  is  also  strengthened,  and  the  syllogism  is 
converted  into  a  universal :  having  the  middle  term  strengthened, 
the  conclusion  is  not  strengthened,  and  the  syllogism  is  converted 
into  a  strengthened  particular  syllogism.      Thus  if  ()  )),  with 
conclusion  (),  have  the  premise  ()  strengthened  into  )),  the  syllo- 
gism becomes  ))  ))  and  yields  )).     But  if  ()  be  strengthened  into 
((,  the  syllogism  becomes  ((  ))  and  yields  only  (),  as  before. 

45.  A  universal  conclusion  affirms  two  particulars :  if  either 
of  these  be  substituted  in  the  conclusion"  of  the  universal  syllogism, 
the  syllogism  may  be  called  a  universal  of  weakened  conclusion 
or  a  weakened  universal.    Thus  X))Y))Z,  made  to  yield  only  X(  )Z 
or  X)(Z,  instead  of  X))Z,  is  a  universal  of  weakened  conclusion. 
No  further  notice  need  be  taken  of  this  case. 

46.  Of  the  24  syllogisms  of  particular  conclusion,  the  con- 
clusions are  equally  divided  among  (),  X>  )*)'  anc^  (*(•     ^^e 
following  table  is  one  of  many  modes   of  arrangement  of  the 









Universal       ^Particular 





X  C) 
(•)  X 

))  X 
X  (( 



The  middle  column  contains  the  universals :  and  each  universal 
stands  horizontally  between  the  two  particulars  into  which  it  may 
be  weakened,  by  weakening  one  of  the  concluding  terms.  And 
each  strengthened  particular  stands  vertically  between  the  two 
particulars  from  which  it  may  be  formed  by  altering  the  quantity 
of  the  middle  term  in  the  particular  premise  only. 

47.  If  two  propositions  give  a  third,  say  A  and  B  give  C ; 
then,  a,  b,  c,  meaning  the  contrary  propositions  of  A,  B,  C,  it 
follows  that  A,  B,  c,  cannot  all  be  true  together.     Hence  if  A,  c, 
be  true,  B  must  be  false,  or  b  true :  that  is,  if  A,  B,  give  C,  then 
A,  c,  give  b.     Or,  either  premise  joined  with  the  contrary  of  the 
conclusion,  gives  the  contrary  of  the  other  premise.     And  thus 
each  form  of  syllogism  has  two  opponent  forms .     But  the  order  of 
terms  will  not  be  correct,  unless  the  premise  which  is  retained  be 
converted.     If  the  order  of  the  terms  in  the  syllogism  be  XY 
YZ  XZ,  we  shall  have  in  one  opponent  XY  XZ  YZ,  which  in 
our  mode  of  arrangement  must  be  YX  XZ  YZ,  the  retained 
premise  changing  the  order  of  its  terms. 

Thus  the  opponent  forms  of  )•)  )),  which  gives  )•),  are  as 
follows.  First,  (•(,  retained  premise  converted;  ((,  contrary  of 
conclusion;  (•(,  contrary  of  other  premise;  giving  (•(  ((  and  con- 
clusion (•(.  Secondly,  ((,  contrary  of  conclusion;  ((,  retained 
premise  converted ;  (( ,  contrary  of  other  premise ;  giving  ((  (( 
and  conclusion  ((. 

48.  The  universal  and  particular  syllogisms  can  be  grouped 
by  threes,  each  one  of  any  three  having  the  other  two  for  its 


opponents.     And  these  groups  can  be  collected  in  the  following 
zodiac,  as  it  may  be  called. 

))  J 

O    w    V:^ 

The  universal  propositions  at  the  cardinal  points  are  so  placed  that 
any  two  contiguous  give  a  universal  syllogism,  whether  read 
forwards  or  backwards,  as  )•(  ((,  ))  )•(.  Join  each  of  these 
universals  with  its  contiguous  external  particular,  so  as  to  read  in 
a  contrary  direction  to  that  in  which  the  two  universals  were  read, 
and  a  triad  is  formed  each  member  of  which  has  the  other  two 
members  for  its  opponent  forms.  As  in 

>(((     >(()     ()));   or  as  in  (•)  ))     X  ((     (OX- 

49.  The  strengthened  particulars  have  weakened  universals 
(§  45)  for  their  opponent  forms.     Thus  ((  ))  with  the  conclusion 
()  has  ))  )•(  with  the  conclusion  (•(  and  )•(  ((  with  the  conclusion 
)•),  for  its  opponents.     And  ((  ((  with  the  conclusion  ()  has  ))  )•( 
with  the  conclusion  ).)  and  )•(  ))  with  the  conclusion  )•)  for  its 

50.  The  partial  terms  of  the  conclusion  take  quantity  in  the 
following  manner, — 

In  universal  syllogisms.  If  one  term  of  the  conclusion  be 
partial,  its  quantity  is  that  of  the  other  term :  if  both,  one  has  at 
least  the  quantity  of  the  whole  middle  term,  and  the  other  of  the 
whole  contrary  of  the  middle  term. 

51.  In  fundamental  particular  syllogisms.     The  partial  term  or 
terms  of  the  conclusion  take  quantity  from  the  particular  premise. 

52.  In  strengthened  particular  syllogisms.     The  partial  term  or 
terms  take  quantity  from  the  whole  middle  term  or  its  whole 
contrary,  according  to  which  is  universal  in  both  of  the  premises. 

53.  These  rules  run  through  every  form  of  the  conclusion  in 
which  there  is  a  particular  term.     Thus  X))Y))Z  gives 

1.  X))Z  in  which  Z  has  the  quantity  of  X 

2.  x((z  in  which  x  has  the  quantity  of  z 

3.  x(-)Z,  xs  as  many  as  ys,  and  Zs  as  many  as  Ys 


Again,  X(-)Y)(Z  gives  X(-(Z,  X()z,  and  x)-)z,  in  which  the 
quantities  of  X(  and  of  )z  are  the  number  of  instances  in  the 
'  some  things '  of  Y)(Z. 

Thirdly,  X>(Y>(Z  gives  X)(Z,  x(-(Z,  x()z,  and  X»,  in 
which  the  quantities  of  x(  and  )z  are  the  number  of  instances 
in  Y. 

54.  A  sorites  is  a  collection  of  propositions  in  which  the  major 
term  of  each  is  the  minor  term  of  the  next,  as  in 


or  All  Xs  are  Ys,  and  No'Y  is  Z,  and  everything  is  either  Z  or 
T,  and  every  T  is  TJ,  and  No  U  is  V,  and  Some  Vs  are  all  Ws. 

55.  A  sorites  gives  a  valid  inference,  1.  Universal,  when  all 
the  premises  are  universal,  and  each  intermediate  term  enters  once 
totally  and  once  partially ;  2.  Particular,  when  one  (and  one  only) 
of  the  two  conditions  just  named  is  broken  once,  whether  by  con- 
tiguous universals  having  an  intermediate  of  one  quantity  in  both, 
or  by  occurrence  of  one  particular  without  breach  of  the  rule  of 

56.  The  inference  is  obtained  by  erasing  all  the  intermediate 
terms  and  their  quantities,  and  allowing  an  even  number  of  dots 
to  indicate  affirmation,  and  an  odd  number  of  dots  to  indicate 

Thus  X))Y>(Z()T>(U(-)V))W  gives  X)-)W 
X(-)Y((Z((T(.)U)(V((W  gives  X(-(W 
X((Y(.)Z>(T(-)U>(V  gives  X((V 

57.  We  have  seen  that  each  universal  may  coexist  with  either 
the  universal  of  altered  quantities  or  with  its  contrary :  which  is 
a  species  of  terminal  ambiguity.     Thus  X))Y  may  have  either 
X((Y  or  X)-)Y  true  at  the  same  time.     All  these  coexistences 
may  be  arranged  and  symbolised  as  follows ;  giving  propositions 
which,  with  reference  to  the  ambiguity  aforesaid,  have  terminal 

1.  X)°)Y  or  both  X))Y  and  X»Y     All  Xs  and  some  things  besides  are  Ys 

2.  X|  |  Y    or  both  X))Y  and  X((Y     All  Xs  are  Ys,  and  all  Ys  are  Xs 

3.  X(°(Y  or  both  X((Y  and  X(-(Y    Among  Xs  are  all  the  Ys  and  some 

things  besides 

4.  X)o(Y  or  both  X)-(Y  and  X)(Y     Nothing  both  X  and  Y   and  some 

things  neither 

5.  X|-|Y  or  both  X)-(Y  and  X(-)Y    Nothing  both  X  and  Y  and  every 

thing  one  or  the  other 

6.  X(o)Y  or  both  X(-)Y  and  X()Y    Every  thing  either  X  or  Y  and  some 

things  both. 




58.  If  any  two  be  joined,  each  of  which  is  1,  3,  4,  or  6,  with 
the  middle  term  of  different  quantities,  these  premises  yield  a 
conclusion  of  the  same  kind,  obtained  by  erasing  the  symbols  of 
the  middle  term  and  one  of  the  symbols  {o}.     Thus  X)o(Y(o)Z 
gives  X)o)Z :  or  if  nothing  be  both  X  and  Y  and  some  things 
neither,  and  if  every  thing  be  either  Y  or  Z  and  some  things  both, 
it  follows  that  all  Xs  and  two  lots  of  other  things  are  Zs. 

59.  In  any  one  of  these  syllogisms,  it  follows  that  ||  may  be 
written  for  )o)  or  (°(  in  one  place,  or  |*j  for  either  )°(  or  (o)  in  one 
place,  without  any  alteration  of  the  conclusion,  except  reducing 
the  two  lots  to  one.     But  if  this  be  done  in  both  places,  the  con- 
clusion is  reduced  to  ||  or  |-|,  and  both  lots  disappear.     Let  the 
reader  examine  for  himself  the  cases  in  which  one  of  the  premises 
is  cut  down  to  a  simple  universal. 

60.  The    rules    of    contraversion    remain    unaltered:    thus 
X(o)Y)o(Z  is  the  same  as  X(o(y(o(Z  &c. 

61.  The  following  exercises  will  exemplify  what   precedes. 
Letters  written  under  one  another  are  names  of  the  same  object. 
Here  is  a  universe  of  12  instances  of  which  3  are  Xs  and  the 
remainder  Ps ;  5  are  Ys  and  the  remainder  Qs ;  7  are  Zs  and 
the  remainder  Rs. 

P  P  P  .P  P 
Q  Q  Q  Q  Q 
R  R  R  R  R 

We  can  thus  verify  the  eight  complex  syllogisms 
X)')Y)°)Z        P(°)Y)°)Z         P(=(Q(°)Z        P( 


P    P 

P    P 

Y  Y  Y 

Y  Y 

Q  Q 

Z    Z    Z 

Z    Z 

Z    Z 

In  every  case  it  will  be  seen  that  the  two  lots  in  the  middle  form 
the  quantity  of  the  particular  proposition  of  the  conclusion. 

62.  The  contraries  of  the  complex  propositions  are  as  follows : 





X|  |Y 



Both  X))Y  and  X)-)Y 

X))Y    -  X((Y 

X((Y    -  X(-(Y 

X)-(Y  -  X)(Y 

X)-(Y  -  X(-)Y 


X(-(Y  or  X((Y  or  both 

X(-(Y  -  X)-)Y  

X)-)Y  -  X))Y    

X()Y  -  X(-)Y  

X()Y  -  X)(Y    

X)(Y    -  X)-(Y  



63.  The  propositions  hitherto  enunciated  are  cumular :  each 
one  is  a  collection  of  individual  propositions,  or  of  propositions 
about  individuals;  X.)~)Y  is  '  All  Xs  are  some  Ys '.  This  pro- 
position is  an  aggregate  of  singular  propositions. 


64.  There  is  a  choice  between  this  cumular  mode  of  con- 
ception and  one  which  may*  be  called  exemplar  •  in  which  each 
proposition  is  the  premise  of  a  unit-syllogism:  as  'this  X  is  one 
Y ',  f  this  X  is  not  any  Y '.  The  distinction  is  seen  in  '  A II  men 
are  animals  '  and  '  Every  man  is  an  animal ',  propositions  of  the 
same  import,  of  which  the  first  sums  up,  the  second  tells  off 
instance  by  instance.  In  the  second,  every  is  synonymous  with 
each  and  with  any. 

*  The  late  Sir  William  Hamilton  entertained  the  idea  of  completing  the 
system  of  enunciation  by  making  the  words  all  (or  when  grammatically 
necessary,  any)  and  some  do  every  kind  of  duty.  He  thus  put  forward,  as 
the  system,  the  following  collection  :  — 


1.  All     X  is  All    Y  

2.  All     X  is  some  Y  X))Y 

3.  Some  X  is  all      Y  X((Y 

4.  Some  X  is  some  Y  XQY 


5.  Any    X  is  not  any    Y  X)'(Y 

6.  Any    X  is  not  some  Y  X)-)Y 

7.  Some  X  is  not  any    Y  X(-(Y 

8.  Some  X  is  not  some  Y  

Of  the  two  propositions  which  are  not  in  the  common  system  (1  and  8) 
the  first  (§  14,  note  f)  is  X|  | Y,  compounded  of  X))Y  and  X((Y :  it  is 
contradicted  by  X(-(Y  and  X)')Y,  either  or  both.  The  second  (8)  is  true 
in  all  cases  in  which  either  X  or  Y  has  two  or  more  instances  in  existence  : 
its  contrary  is  '  X  and  Y  are  singular  and  identical ;  there  is  but  one  X, 
there  is  but  one  Y,  and  X  is  Y '.  A  system  of  propositions  which  mixes  the 
simple  and  the  complex,  which  compounds  two  of  its  own  set  to  make  a  third 
in  one  case  and  one  only,  §  57,  and  which  offers  an  assertion  and  denial 
which  cannot,  be  contradicted  in  the  system,  seems  to  me  to  carry  its  own 
condemnation  written  on  its  own  forehead.  From  this  system  I  was  led  to 
the  exemplar  system  in  the  text.  For  Sir  W.  Hamilton's  defence  of  his  own 
views,  and  objections  to  mine,  see  his  Discussions  on  Philosophy,  &c.  Appen- 
dix B.  In  making  this  reference,  however,  it  is  due  to  myself  to  warn  the 
reader  who  has  not  access  to  the  paper  criticised  that  Sir  W.  Hamilton  did 
not  read  with  sufficient  attention,  partly  no  doubt  from  ill  health.  The 
consequence  is  that  I  must  not  be  held  answerable  for  all  that  is  represented 
by  him  as  coming  from  me.  For  example,  speaking  of  my  Table  of  exemplar 
propositions,  he  says  "  And  mark  in  what  terms  it  [the  table  of  exemplars] 
is  ushered  in : — as  '  a  system  . . . .'  Nay,  so  lucid  does  it  seem  to  its  inventor, 
that,  after  the  notation  is  detailed,  we  are  told  that  it '  needs  no  explanation?  " 
The  paragraph  here  criticised  had  two  notations,  one  of  which  I  called  the 
detailed  notation,  because  there  is  more  detail  in  it  than  in  the  other :  the 
other  is  the  old  notation,  augmented.  The  first  had  been  sufficiently  ex- 
plained in  what  preceded  ;  the  second  was,  as  to  the  augmentations,  new  to 
the  reader.  Accordingly,  the  table  being  finished,  I  proceeded  thus  "  The 
detailed  notation  needs  no  explanation.  The  form  given  to  the  old  notation 

may  be  explained  thus "     Sir  W.  Hamilton  represented  me  as  saying 

that  after  the  notation  [all  the  notation,  I  suppose]  is  detailed,  it  [table  or 
notation,  I  know  not  which]  needs  no  explanation.     I  select  this  small  point 


as  one  that  can  be  briefly  dealt  with  :  there  are  many  more,  which  I  shall 
probably  never  notice,  unless  it  be  one  at  a  time  as  occasion  of  illustration 
arises.  A  very  decisive  case  is  exposed  in  the  postscript  of  my  third  paper  in 
the  Cambridge  Transactions. 

65.  Quantity  is  now  replaced  by  mode  of  selection.     There  is 
unlimited  selection,  expressed  by  the  word  any  one :  vaguely  limited 
selection,  expressed  by  some  one.     When  we  say  some  one  we 
mean  that  we  do  not  know  it  may  be  any  one. 

66.  Let  (X  and  X)  now  mean  any  one  X:  let  )X  and  X( 
mean  some  one  X. 

67.  The  propositions  are  as  follows :    the  first  of  each  pair 
being  a  universal,  the  second  its  contrary  particular. 

Exemplar  form.  Cumular  form. 

X)(Y    Any  one  X  is  any  one  Y  X  and  Y  singular  and  identical 

X(-)Y  Some  one  X  is  not  some  one  EitherX  not  singular,  or  Y  not  singu- 
Y  lar ;  or  if  both  singular,  not  identical 

X))Y    Any  one  X  is  some  one  Y  All  Xs  are  some  Ys 

X(-  (Y  Some  one  X  is  not  any  one  Y  Some  Xs  are  not  (all)  Ys 

X((Y    Some  one  X  is  any  one  Y  Some  Xs  are  all  Ys 

X)  •)  Y  Any  one  X  is  not  some  one  Y      All  Xs  are  not  some  Ys 

X)-(Y  Any  one  X  is  not  any  one  Y      All  Xs  are  not  (all)  Ys 
X()Y  Some  one  X  is  some  one  Y         Some  Xs  are  some  Ys. 

Six  of  the  forms  of  this  exemplar  system  are  identical  with  six 
of  the  forms  of  the  cumular  system.  And  these  six  forms  are  the 
forms  of  the  old  logic,  if  we  take  care  always  to  read  X((Y  and 
X)-)Y  backwards,  and  to  count  X)-(Y  and  X()Y  as  each  a 
pair  of  propositions,  by  distinguishing  the  reading  forwards  from 
the  reading  backwards. 

68.  The  two  new  forms  of  the  exemplar  system  (the  first  and 
second  above)  come  under   the   same  symbols  as  the  two  new 
forms  of  the  cumular  system,  (•)  and  )(:  but  the  meanings  are 
widely  different.     Both  systems  contain  every  possible   combi- 
nation of  quantities,  as  well  in  universal  as  in  particular   pro- 

69.  If  the  above  propositions  be  applied  to  contraries,  we  have 
a  more  extensive  system  of  propositions.     I  shall  not  enter  on  this 
enlargement,  because   the  peculiar  proposition   of  this   system, 
X)(Y,  is  of  infrequent*  use  in  thought  as  connected  with  the 
consideration  of  X  and  Y  in  opposition  to  their  contraries. 

*  All  necessary  laws  of  thought  are  part  of  the  subject  of  logic  :  but  a 
small  syllabus  cannot  contain  everything.  The  rejection  from  logic,  and  the 
rejection  from  a  book  of  logic,  are  two  very  different  things.  It  has  not  been 


26  EXEMPLAR  SYLLOGISM.  [69-71. 

uncommon  to  repudiate  rare  and  unusual  forms  from  the  science  itself,  by 
calling  them  subtleties,  or  the  like.  This  (§  73)  is  not  reasonable :  but  as 
to  the  contents  of  a  work,  especially  of  a  syllabus,  the  time  must  come  at 
which  any  one  who  asks  for  more  Inust  be  answered  by 

Cum  tibi  sufficiant  cyathi,  cur  dolia  quaeris  ? 

As  another  example  : — I  have,  §  16,  required  that  no  term  shall  be  intro- 
duced which  fills  the  whole  universe.  In  common  logic,  with  an  unlimited 
universe,  there  is  really  no  name  as  extensive  as  the  universe  except  object 
of  thought.  But  it  is  otherwise  in  the  limited  universes  which  I  suppose. 
A  short  and  easy  chapter  on  names  as  extensive  as  the  universe  might  be 
needed  in  a  full  work  on  logic,  but  not  in  a  syllabus. 

70.  To  make  a  syllogism  of  valid  inference,  it  is  enough  that 
there  be  at  least  one  unlimited  selection  of  the  middle  term,  and 
at  least  one  affirmative  proposition.     And  the  inference  is  obtained 
by  dropping  all  the  symbols  of  the  middle  term.     Thus  X((Y(-)Z 
shows  premises  which  give  the  conclusion  X(')Z:  or  '  Some  one 
X  is  any  one  Y  and  Some  one  Y  is  not  some  one  Z  '  giving  '  Some 
one  X  is  not  some  one  Z  '. 

71.  There  are  36  valid*  forms  of  syllogism,  as  follows,  read- 
ing each  symbol  both  backwards  and  forwards,  but  not  counting 
it  twice  when  it  reads  backwards  and  forwards  the  same,  as  in 

XX,  (())• 

Fifteen  in  which  X  is  joined  with  itself  or  another, — 

XX   )())    )(((    XX    )()•)    )((*(   )(()   )((') 

Fifteen  in  which  the  syllogism  is  but  an  exemplar  reading  of 
a  cumular  syllogism, — 

))))   0))   (())   ))>(.  ((>(   OX    )))')   )'))) 

Six  which  give  the  conclusion  (•), — 

(((')      (OO       0)0 

*  If  Sir  William  Hamilton's  system  be  taken,  there  are  also  36  valid 
forms  of  syllogism,  the  same  as  in  the  text :  but  the  law  of  inference  is 
slightly  modified,  as  follows.  When  both  the  middle  spicula?  turn  one  way, 
as  in  ))  and  ((,  then  any  spicula  of  universal  quantity  which  turns  the  other 
way  must  itself  be  turned,  unless  it  be  protected  by  a  negative  point.  Thus 
)(  (),  which  in  the  exemplar  system  gives  )),  in  the  cumular  system 
gives  (). 

Exemplar  system. 
Any  one  X  is  any  one  Y 
Some  one  Y  is  some  one  Z 
Therefore  Any  one  X  is  some  one  Z 

Cumular  system. 
All  Xs  are  all  Ys. 
Some  Ys  are  some  Zs. 
Therefore  Some  Xs  are  some  Zs. 

This  distinction  will  afford  useful  study.     The  minor  premise  of  the  exem- 
plar instance  implies  that  there  is  but  one  X  and  one  Y. 


72.  The  exemplar  proposition  is  not  unknown.     It  is  of  very 
frequent  use  in  complete  demonstration.     When  Euclid  proves 
that  Every  triangle  has  angles  together  equal  to  two  right  angles, 
he  selects,  or  allows  his  reader  to  select,  a  triangle,  and  shows 
that  any  triangle  has  -angles  equal  to  two  right  angles :  and  the 
force  of  demonstration  is  for  those  who  can  see  that  the  selection* 
is  not  limited  by  anything  in  the  reasoning.     The  exemplar  form 
of  enunciation,  then,  is  of  at  least  as  frequent   use  in   purely 
deductive  reasoning   as  any  other ;    and  is  therefore  fitly  intro- 
duced even  into  a  short  syllabus.     In  any  case  it  is  a  subject  of 
logical  consideration,  as  being  an  actual  fonn  of  thought. 

*  The  limitation  of  the  selection  by  some  detail  of  process  is  one  of  the 
errors  against  which  the  geometer  has  especially  to  guard.     I  remember  an 
asserted  trisection  of  the  angle  which  I  examined  again  and  again  and  again, 
without  being  able  to  detect  a  single  offence  against  Euclid's  conditions.     At 
last,  in  the  details  of  a  very  complex  construction,  I  found  two  requirements 
which  were  only  possible  togtther  on  the  supposition  of  a  certain  triangle 
having  its  vertex  upon  the  base.     Now  it  happened  that  one  of  the  angles 
at  the  base  of  this  triangle  was  the  very  angle  to  be  trisected :  so  that  the 
author  had  indeed  trisected  an  angle,  but  not  any  angle ;  he  had  most  satis- 
factorily, and  by  no  help  but  Euclid's  geometry,  divided  the  angle  0  into 
three  equal  parts,  0,  0,  0.     A  modification  of  his  process  would  have  been 
equally  successful  with  180°,  which  Euclid  himself  had  trisected. 

73.  The  following  passage,  written  by  Sir  Wrlliam  Hamilton 
himself,  should  be  quoted  in  every  logical  treatise:  for  it  ought 
to  be  said,  and  cannot  be  said  better.     "  Whatever  is  operative  in 
thought,  must  be  taken  into  account,  and  consequently  be  overtly 
expressible  in  logic ;  for  logic  must  be,  as  to  be  it  professes,  an 
unexclusive  reflex  of  thought,  and  not  merely  an  arbitrary  selec- 
tion—  a  series  of  elegant  extracts- — out  of  the  forms  of  thinking. 
Whether  the  form  that  it  exhibits  as  legitimate  be  stronger  or 
weaker,  be  more  or  less  frequently  applied;  —  that,  as  a  material 
and  contingent  consideration,  is  beyond  its  purview." 

74.  The  heads*  of  the  numerically  definite  proposition  and 
syllogism  are  as  follows, — 

Let  u  be  the  whole  number  of  individuals  in  the  universe. 
Let  x,  y, z, be  the  numbers  of  Xs,  Ys,  and  Zs.  Then  u—  x,  u—y, 
u — z  are  the  numbers  of  xs,  ys,  and  zs. 

*  On  this  subject  I  have  given  only  heads  of  result,  the  demonstrations 
of  which  will  be  found  in  my  Formal  Logic. 

.  75.  Let  mXY  mean  that  m  or  more  Xs  are  Ys.  Then  mXy 
means  that  m  or  more  Xs  are  ys,  or  not  Ys.  And  wiYX  and 
myX  have  the  same  meanings  as  TnXY  and  wXy. 


76.  Let  a  proposition  be  called  spurious  when   it  must  of 
necessity  be  true,  by  the  constitution  of  the  universe.     Thus,  in  a 
universe  of  100  instances,  of  which  70  are  Xs  and  50  are  Ys,  the 
proposition  20  XY  is  spurious :  for  at  least  20  Xs  must  be  Ys, 
and  20  XY  cannot  be  denied,  and  need  not  be  affirmed  as  that 
which  might  be  denied. 

77.  Let  every  negative  quantity  be  interpreted  as  0 :    thus 
(6-10)  XY  means  that  none  or  more  Xs  are  Ys,  a  spurious  pro- 

78.  The  quantification  of  the  predicate  is  useless.     To  say 
that  mXs  are  to  be  found  among  nYs,  is  no  more  than  is  said  in 
raXY.    To  say  that  raXs  are  not  any  one  to  be  found  among  any 
lot  of  nYs  is  a  spurious  proposition,  unless  m  +  n  be  greater  than 
both  x  and  y,  in  which  case  it  is  merely  equivalent  to  both  of  the 
following,  (m  +  n — y)  Xy,   and  (m  +  n — #)  Yx,  which  are  equi- 
valent to  each  other. 

79.  In  raXY,  the   spurious  part,  if  any,  is  (x  +  y — w)XY; 
the  part  which  is  not  spurious  is  (m  +  u — x — ?/)XY.     For  each 
instance  in  the  last  there  must  be  an  x  which  is  y.     The  follow- 
ing pairs  of  propositions  are  identical. 

m  XY  and  (m+u — x — y)  xy 
mXy    and  (m+y— T)  xY 
m  xY    and  (m  +x — y)  Xy 
wixy     and  (m+x+y — M)  XY 


Their  contraries. 

(x+l-m)  Xy  (y+l_m)  xY 

(x+l  — TW)  XY  (u+l—y-m)xy 

(u+l  —  ar-m)  xy          (y  +  l-wi)  XY 

Identical  propositions. 
TO  XY       (m+u— x— y)  xy 
m  Xy        (m+y — x)  xY 
m  xY         (m+x — y)  Xy 
77i  xy          (m+x+y — u)  XY 

81.  From  mXY  and  wYZ  we  infer  (in  +  n—y)  XZ,  or  its 
equivalent  (m  +  n  +  u — a: — y — z)xz.  The  four  following  forms 
include  all  the  cases  of  syllogism :  the  first  two  columns  show  the 
premises,  the  second  two  the  identical  conclusions, — 

m  XY  n  YZ  (m+n— y)  XZ  (m+n+u— x—y—z)  xz 

m  Xy  7i  YZ  (m+n— x)  xZ  (m+n-z)  Xz 

m  XY  n  yZ  (m+n— z)  Xz  (m+n— x)  xZ 

.    m  Xy  n  yZ  (m+n+y— x— z)  xz  (m+n+y— w)XZ 

82.  When  either  of  the  concluding  terms  is  changed  into  its 
contrary,  the  corresponding  changes  are  made  in  the  forms  of 
inference.  Thus  to  find  the  inference  from  mxy  and  wyz,  we 


must,  in  the  fourth  form,  write  x  for  X,  z  for  Z,  X  for  x,  Z  for  z, 
u — x  for  x,  and  u — z  for  z. 

83.  A  spurious  premise  gives  a  spurious  conclusion:    and 
premises  neither  of  which  is  spurious  may  give  a  spurious  con- 
clusion.    A  proposition  is   only  spurious  as  it   is  known  to  be 
spurious :  hence  when  u,  x,  y,  z  are  not  known,  there  are  no 
spurious  propositions. 

84.  Every  proposition  has  two  forms,  one  of  names  contrary 
to  the  other,  both  spurious,  or  neither.     Whenever  X()Y  is  true 
in  a  manner  which,  by  the  constitution  of  the  universe,  might 
have  been  false,  then  x()y,  or  X)(Y  is  also  true  in  the  same 
manner.     The  ordinary  syllogism  would  have  two  such  contra- 
nominal  forms  of  one  conclusion,  and,  properly  speaking,  has  two 
such  forms.     When  the  conclusion  is  universal,  we  know  it  has 
them:  for  X))Z  is  x((z,  X)-(Z  is  x(')z,  &c.     These  we  may  see 
to  be  the  contranominal  conclusions  of  the  numerical  syllogism. 
For  X))Y  is  #XY,  and  Y))Z  is  yYZ,  whence  O+?/-r/)XZ  and 
(x+y— y  +  u — x— ^)xz,  or  #XZ,  which'is  X))Z,  and  (u — z)  xz, 
which  is  x((z.      Again,  let   X()Y  be  wXY,  then,  Y))Z  being 
?/YZ,  we  have  (m+y— ?/)XZ,  or  mXZ,  and  (m  +  u— x— z)  xz, 
its  equivalent.     If  x,  z,  u,  be  known,  then  if  m  XZ  be  any  thing 
except  what  must  be,  we  have  m  +  u~>x  +  z,  and  (m  +  u—x — z)  xz 
is  x(  )z  or  X)(Z.     As  it  is,  x,  y}  u,  being  unknown,  we  have  raXZ 
certainly  true,  be  it  spurious  or  not,  and  we  can  say  nothing 
of  (m+u — x — z) xz. 

85 .  Syllogisms  with  numerically  definite  quantity  rarely  occur, 
if  ever,  in  common  thought.     But  syllogisms  of  transposed  quan- 
tity occur,  in  which  the  number  of  instances  of  one  term  is  the 
whole  possible  number  of  instances  of  another  term.    For  example; 
— cFor  every   Z  there  is  an  X  which  is  Y;    some   Zs  are  not 
Ys'.     Here  we  have  zXY  and  wyZ  ;  whence  (z  +  n — ^)Xz  and 
(z-\-n — #)xZ.    The  first  is  wXz,  a  case  of  X('(Z  ;  some  Xs  are 
not  Zs.     Thus,  '  For  every  man  in  the  house  there  is  a  person 
who  is  aged;  some  of  the  men  are  not  aged':    it  follows,  and 
easily,  that  some  persons  in  the  house  are  not  men ;  but  not  by 
any  common  form  of  syllogism. 

86.  Of  terms  in  common  use  the  only -one  which  can  give  the 
syllogisms  of  this  chapter  is  e  most '.     As  in 

Most  Ys  are  Xs ;  most  Ys  are  Zs ;  therefore  some  Xs  are  Zs. 
Most  Ys  are  Xs ;  most  Ys  are  not  Zs ;  therefore  some  Xs  are 
not  Zs. 



Most  Ys  are  not  Xs;  most  Ys  are  not  Zs;  therefore  some 
things  are  neither  Xs  nor  Zs. 

87.  Each  one  of  our  syllogisms  may  be  stated  in  eight 
different  ways,  each  premise  and  the  conclusion  admitting  two 
different  orders.  Thus  X))Y,  Y))Z,  giving  X))Z  may  be  stated 
as  Y((X  Y))Z  giving  Z((X,  or  as  X))Y,  Y))Z,  giving  Z((X,  &c. 
All  the  orders  are  as  follows  — 









88.  Whenever  there  is  a  first  and  a  second,  let  them  be  called 
minor  and  major.  Write  the  premises  so  that  the  minor  premise 
shall  contain  the  minor  term  of  the  conclusion  (though  it  has 
long  been  most  common  to  write  the  major  premise  first),  and 
we  have 








These  orders  are  called  the  four  figures.  Thus  X))Y  Y))Z 
giving  X))Z  is  stated  in  the  first  figure;  X))Y  Z((Y  giving 
X))Z  is  stated  in  the  second  figure ;  Y((X  Y))Z  giving  X))Z  is 
stated  in  the  third  figure ;  Y((X  Z((Y  giving  X))Z  is  stated  in 
the  fourth  figure. 

89.  The  .first  figure  may  be  called  the  figure  of  direct  transi- 
tion :  the  fourth,  which  is  nothing  but  the  first  with  a  converted 
conclusion,  the  figure  of  inverted  transition ;  the  second,  the  figure 
of  reference  to  (the  middle  term) ;  the  third,  the  figure  of  reference 
from  (the  middle  term). 

90.  The  first  figure  is  the  one  which  has  been  used  in  our 
symbols;    and  it  is   the   most   convenient.      The  distinction   of 
figure  is   wholly  useless    in  this    tract,    so    far  as  we  have  yet 
gone:  it  becomes  necessary  when  we  take  a  wider  view  of  the 

91.  A  convertible  copula  is  one  in  which  the  copular  relation 
exists  between  two  names  both  ways :  thus  '  is  fastened  to '  '  is 
joined  by  a  road  with  '  '  is  equal  to, '  '  is  in  habit  of  conversation 
with/  &c.  are  convertible  copulse.     If  '  X  is  equal  to  Y '  then  '  Y 
is  equal  to  X '  &c. 

92.  A  transitive*  copula  is  one  in  which  the  copular  relation 
joins  X  with  Z  whenever  it  joins  X  with  Y  and  Y  with  Z.     Thus 
*  is  fastened  to '  is  usually  understood  as  a  transitive  copula :  '  X 

92-93.]  EXTENSION  OF  COPULA.  31 

is  fastened  to  Y '  and  '  Y  is  fastened  to  Z  '  give  '  X  is  fastened 
to  Z'. 

*  All  the  copulas  used  in  this  syllabus  are  transitive.  The  intransitive 
copula  cannot  be  treated  without  more  extensive 'consideration  of  the  combi- 
nation of  relations  than  I  have  now  opportunity  to  give  :  a  second  part  of 
this  syllabus,  or  an  augmented  edition,  may  contain  something  on  this 

93.  The  junction  of  names  by  appiirtenance  to  one  object,  the 
copula  hitherto  used,  is  both  convertible  and  transitive :  and  from 
these  qualities,  and  from  these  alone,  it  derives  the  whole  of  its 
functional  power  in  syllogism.  Any  copula  which  is  both  transi- 
tive and  convertible  will  give  precisely  the  syllogisms*  of  our 
system,  and  no  others :  provided  always  that  if  contrary  names 
be  introduced,  no  instance  of  a  name  can,  either  directly  or  by 
transition,  be  joined  by  the  copula  with  any  instance  of  the  con- 
trary name.  For  example,  let  the  copula  be  some  transitive  and 
convertible  mode  of  joining  or  fastening  together,  whether  of 
objects  in  space  or  notions  in  the  rnind,  &c. :  so  that  no  X  is  ever 
joined  with  any  x,  &c.  The  following  are  two  instances  of 
syllogism.  • 

X))Y)'(Z.  Every  X  is  joined  to  a  Y;  no  Y  is  joined  to  a  Z  ; 
therefore  no  X  is  joined  to  a  Z.  For  if  any  X  were  joined  to  a  Z, 
that  Z  would  be  joined  to  an  X,  and  that  X  to  a  Y,  whence  that 
Z  would  be  joined  to  a  Y,  which  no  Z  is. 

X(-)Y)(Z.  Everything  is  joined  either  to  an  X  or  to  a  Y; 
Some  things  are  joined  neither  to  Ys  nor  to  Zs ;  therefore  Some 
Xs  are  not  joined  to  Zs.  For  if  every  X  were  joined  to  a  Z, 
then  every  thing  being  (by  the  first  premise)  joined  either  to  an 
X  or  to  a  Y,  is  joined  either  to  a  Z  or  to  a  Y,  which  contradicts 
the  second  premise. 

*  The  logicians  are  aware  that  many  cases  exist  in  which  inference  about 
two  terras  by  comparison  with  a  third  is  not  reducible  to  their  syllogism. 
As  '  A  equals  B  ;  B  equals  C ;  therefore  A  equals  C.'  This  is  not  an  instance 
of  common  syllogism  :  the  premises  are  '  A  is  an  equal  of  B  ;  B  is  an  equal 
of  C.'  So  far  as  common  syllogism  is  concerned,  that  'an  equal  of  B' 
is  as  good  for  the  argument  as  'B'  is  a  material  accident  of  the  meaning  of 
'  equal.'  The  logicians  accordingly,  to  reduce  this  to  a  common  syllogism, 
state  the  effect  of  composition  of  relation  in  a  major  premise,  and  declare 
that  the  case  before  them  is  an  example  of  that  composition  in  a  minor 
premise.  As  in,  A  is  an  equal  of  an  equal  (of  C)  :  Every  equal  of  an  equal  is 
an  equal ;  therefore  A  is  an  equal  of  C.  This  I  treat  as  a  mere  evasion. 
Among  various  sufficient  answers  this  one  is  enough  :  men  do  not  think  as 
above.  When  A  =  B,  B  =  C,  is  made  to  give  A  =  C,  the  word  equals  is  a 

32  EXTENSION  OF  COPULA.  [93-97. 

copula  in  thought,  and  not  a  notion  attaeJied  to  a  predicate.  There  are 
processes  which  are  not  those  of  common  syllogism  in  the  logician's  major 
premise  above :  but  waiving  this,  logic  is  an  analysis  of  the  form  of  thought, 
possible  and  actual,  and  the  logician  has  no  right  to  declare  that  other  than 
the  actual  is  actual. 

94.  The  convertibility  of  the  copula  renders   the  inference 
altogether  independent  of  figure. 

95.  Let  the  copula  be  inconvertible,  as  in  'X  precedes  Y' 
from  which  we  cannot  say  that '  Y  precedes  X '.     We  must  now 
introduce  the  converse  relation  'Y  follows  X',  and  the  conversion 
of  a  proposition  requires  the  introduction  of  the  converse  copula. 

96.  This  extension,  when   contraries  are   also   introduced,  is 
almost  unknown  in  the  common  run  of  thought :  but  it  may  serve 
for  exercise,  and  also  to  give  an  idea  of  one  of  those  innumerable 
systems  of  relation  with  which  thought  unassisted  by  systematic  * 
analysis  would  probably  never  become  familiar. 

*  The  uneducated  acquire  easy  and  accurate  use  of  the  very  simplest 
cases  of  transformation  of  propositions  and  of  syllogism.  The  educated, 
by  a  higher  kind  of  practice,  arrive  at  equally  easy  and  accurate  use  of  some 
more  complicated  cases  :  but  not  of  all  those  which  are  treated  in  ordinary 
logic.  Euclid  may  have  been  ignorant  of  the  identity  of  "  Every  X  is  Y"  and 
"  Every  not-Y  is  not-X,"  for  any  thing  that  appears  in  his  writings  :  he  makes 
the  one  follow  from  the  other  by  new  proof  each  time.  The  followers  of 
Aristotle  worked  Aristotle's  syllogism  into  the  habits  of  the  educated  world, 
giving,  not  indeed  anything  that  demonstrably  could  not  have  been  acquired 
without  system,  but  much  that  very  probably  would  not.  The  modern 
logician  appeals  to  the  existing  state  of  thought  in  proof  of  the  completeness 
of  the  ordinary  system :  he  cannot  see  anything  in  an  extension  except  what 
he  calls  a  subtlety.  In  the  same  manner  a  country  whose  school  of  arith- 
metical teachers  had  never  got  beyond  counting  with  pebbles  would  be  able 
to  bring  powerful  arguments  against  pen,  ink,  and  paper,  the  Arabic 
numerals,  and  the  decimal  system.  They  would  point  to  society  at  large 
getting  on  well  enough  with  pebbles,  and  able  to  do  all  their  work  with  such 
means :  for  it  is  an  ascertained  fact  that  all  which  is  done  by  those  to  whom 
pebbles  are  the  highest  resource,  is  done  either  with  pebbles  or  something 
inferior.  I  have  long  been  of  opinion  that  the  reason  why  common  logic  is 
lightly  thought  of  by  the  mass  of  the  educated  world  is  that  the  educated 
world  has,  in  a  rough  way,  arrived  at  some  use  of  those  higher  developments 
of  thought  which  that  same  common  logic  has  never  taken  into  its  compass. 
Kant  said  that  the  study  of  a  legitimate  subtlety  (necessary  but  infrequent 
law  of  though*,)  sharpens  the  intellect,  but  is  of  no  practical  use.  Sharpen 
the  intellect  with  it  until  it  is  familiar,  and  it  will  then  become  of  practical 
use.  A  law  of  thought,  a  necessary  part  of  the  machinery  of  our  minds,  of 
no  practical  use !  Whose  fault  is  that  ? 

97.  Let  any  two  names  be  connected  by  transitive  converse 
relations,  for  an  example  say  gives  to  and  receives  from  (under- 



standing  that  when  X  gives  to  Y  and  Y  gives  to  Z,  X  gives  to  Z) 
in  the  following  way, — 

No  X  gives  to  another  X,  either  directly  or  transitively,  &c. 
Every  X  either  gives  to  a  Y  or  receives  from  a  y,  but  not  both 
Every  x  either  gives  to  a  Y  or  receives  from  a  y,  but  not  both 
Every  X  which  gives  to  a  Y,  receives  from  no  other  Y,  &c. 

The  same  of  all  combinations  of  names,  as  Y  with  X  and  x,  &c. 

98.  The  following  are  the  propositions  used,  with  their 
symbols ;  and  in  a  corresponding  way  for  any  other  copula  which 
may  be  used, — 

X)')Y   Every  X  gives  to  a  Y 
X('-(Y  Some  Xs  give  to  no  Ys 
X)'-(Y  No  X  gives  to  a  Y 
X(')Y    Some  Xs  give  to  Ys 
X('-)  Y  In  every  relation,  something 
either  gives  to  an  X  or  re- 
ceives from  a  Y  (or  both) 
X)'(Y    In   some  relations,  nothing 
gives  to  any  X  nor  receives 
from  any  Y 

X('(Y   Some  Xs  give  to  all  the  Ys 
X)'-)Y  All  Xs  do  not  give  to  some 

X))'Y   Every  X  receives  from  a  Y 
X(-('Y  Some  Xs  receive  from  no  Ys 
X)-('Y  No  X  receives  from  a  Y 
XQ'Y    Some  Xs  receive  from  Ys 
X(')'Y  In  every  relation,  something 

either  receives  from  an  X 

or  gives  to  a  Y  (or  both) 
X)('Y    In   some  relations,  nothing 

receives  from  any  X  nor 

gives  to  any  Y 

X((TT    Some  Xs  receive  from  all  Ys 
XyyY  All  Xs  do  not  receive  from 

some  Ys 

99.  Propositions  are  changed  into  others  identical  with  them  by 
this  addition  to  the  rule  in  §  26  : — When  one  term  is  contraverted, 
the  relation  is  also  converted:  when  both,  the  relation  remains. 
In  the  following  lists  the  four  in  each  line  are  identical, — 


X)'-(Y  X))'y 

X('-)Y  X(('y 

X('(Y  X(-)'y 







x)('Y  x)'-)y 
x)-)'Y  x)'(y 
x(-CY  x(')y 

The  relations  may  be  converted  throughout. 

100.  To  prove  an  instance,  how  do  we  know  that  X)')Y  is 
identical  with  x(')'Y?  If  every  X  give  to  a  Y,  the  remaining 
Ys,  if  any,  do  not  give  to  any  Xs,  by  the  assigned  conditions  of 
meaning :  consequently  those  remaining  Ys  receive  from  xs.  As 
to  ys,  none  of  them  can  give  to  Xs,  for  then  they  would  give  to 
Ys :  therefore  all  receive  from  xs.  Conversely,  if  x(-/Y,  no  X 
can  receive  from  y,  for  then  neither  could  that  y  receive  from 
x,  nor  could  that  X  give  to  Y :  so  that  there  would  be  a  relation 
in  which  neither  does  any  thing  give  to  Y,  nor  receive  from  x. 
Consequently,  every  X  gives  to  Y. 


101.  Let  the  phases  of  a  figure  depend  on  the  quality  of  the 
premises  in  the  following  manner:    +  meaning  affirmative,  and 
—  negative,  remember  the  phases  in  the  following  order, — 

102.  For  the  four  figures,  let  these  four  phases  be  the  first  or 

primary  phases:   thus   H is  the  primary  phase  of  the   third 

figure.     To  put  the  other  phases  in  order,  read  backwards  from 
the  primary  phases,  and  then  forwards. 

1  2              34 

Figure    I.      +  +  —  +  +  — 

II.     -  +  ++  +  - 

III.      +-  -  +  +  +        -  - 

Thus  +  —  is  the  third  phase  of  the  second  figure. 

103.  In  the  primary  phases,  the  direct  copula  may  be  used 
throughout.     When  one  premise  departs  from  the  primary  phase 
in  quality,  the  converse  copula  must  be  used  in  the  other ;  when 
both,  in  the  conclusion.     This  addition  is  all  that  is  required  in  the 
treatment  of  the  syllogism  of  inconvertible  copula. 

104.  Thus,*  the  premises  being  X)-(Y  Z))Y,  we  have  the 
primary  phase  of  the  second  figure,  whence  X)-(Z  with  the  direct 
copula.     That  is,  if  no  X  give  to  a  Y,  and  every  Z  give  to  a  Y, 
no  X  gives  to  a  Z.     For  if  any  X  gave  to  a  Z,  that  Z  giving  to 
a  Y,  that  X  would  give  to  a  Y,  which  no  X  does.     Now  con- 
travert  the  middle  term,  and  we  have  X))y  Z)-(y,  the  phase  of 
the  second  figure  in  which  both  premises  differ  from  the  primary 
phase.     Hence  Every  X  gives  y,  No  Z  gives  y,  yields  no  X  is 
given  by  Z.     For  if  any  X  were  given  by  Z,  a  y  would  be  given 
by  that  Z,  which  is  given  by  no  Z.     But  'no  X  gives  Z'  will 
not  do. 

*  The  reader  may  exercise  himself  in  the  formation  of  more  examples. 
The  use  of  such  a  developement  as  the  one  before  him  is  this.  Every  study 
of  a  generalisation  or  extension  gives  additional  power  over  the  particular 
form  by  which  the  generalisation  is  suggested.  Nobody  who  has  ever 
returned  to  quadratic  equations  after  the  study  of  equations  of  all  degrees, 
or  who  has  done  the  like,  will  deny  my  assertion  that  Ol  ^i-rti  ftxi^nav  may 
be  predicated  of  any  one  who  studies  a  branch  or  a  case,  without  afterwards 
making  it  part  of  a  larger  whole.  Accordingly,  it  is  always  worth  while  to 
generalise,  were  it  only  to  give  power  over  the  particular.  This  principle,  of 
daily  familiarity  to  the  mathematician,  is  almost  unknown  to  the  logician. 

105.  The  common  system  of  syllogism,  which  being  nearly 
complete  in  the  writings  of  Aristotle  may  be  called  Aristotelian, 


is  as  much  as  may  be  collected  out  of  the  preceding  system  by 
the  following  modifications,  — 

1  .  The  exclusion  of  all  idea  of  a  limited  universe,  of  contrary 
names,  and  of  the  propositions  (•)  and  )(.  2.  The  exclusion  of 
all  right  to  convert  a  proposition,  except  when  its  two  terms  have 
like  quantities,  as  in  )•(  and  ().  Thus  X))Y  must  not  be  read  as 
*  Some  Ys  are  all  Xs  '.  But  X))Y  may  undergo  what  is  called 
the  conversion  per  accidens  :  that  is,  X))Y  affirming  X(  )Y,  which 
is  Y()X,  X))Y  may  be  made  to  give  YQX.  3.  The  exclusion 
of  every  copula  except  the  transitive  and  convertible  copula. 

4.  The  addition  of  the  consideration  of  the  identical  pairs  X)'(Y 
and  Y)-(X,  X()Y  and  Y()X,  as  perfectly  distinct  propositions. 

5.  The  introduction  of  the  distinction  of  figure.     6.  The  writing 
of  the  major  and  minor  propositions  first  and  second,  instead  of 
second  and  first:  thus  X))Y))Z  is  written  'Y))Z,  X))Y,  whence 

106.  There  are  four  forms  of  proposition  :    A,  or  X))Y  or 
Y))X,  not  identical;    E,   or  X>(Y  or  Y)-(X,  identical;    I,   or 
X()Y  or  Y()X,  identical;  O,  or  X(-(Y  or  Y(-(X,  not  identical. 

107.  There  are  four  fundamental  syllogisms  in  the  first  figure, 
each  of  which  has  an  opponent  in  the  second,  and  an  opponent  in 
the  third.     There  are  three  fundamental  syllogisms  in  the  fourth 
figure,  each  of  which  has  the  other  two  for  opponents.     Alto- 
gether, fifteen  fundamental  syllogisms.    There  are  three  strength- 
ened particular  syllogisms,  two  in  the  third  figure,  and  one  in  the 
fourth  :  and  one  weakened  universal,  in  the  fourth  figure.     In  all, 
nineteen  forms. 

108.  Every  syllogism  has  a  word  attached  to  it,  the  vowels  of 
which  are  those  of  its  premises  and  conclusion.    In  the  first  figure 
the  consonants  are  all  unmeaning;  in  the  other  figures  some  of 
the  consonants  give  direction  as  to  the  manner  of  converting  into 
the  first  figure.     Thus  K  denotes  that  the  syllogism  cannot  be 
directly  converted  into  the  first  figure,  though  its  opponent  in  the 
first  figure  may  be  used  to  force  its  conclusion.     S  means  that 
the  premise  whose  vowel  precedes  is  to  be  simply  converted.     P, 
which  occurs  in  all  the  strengthened  particulars  and  the  weakened 
universal,  means  that  the  conversion  per  accidens  is  to  be  employed 
on  the  preceding  member.     M  means  that  the  premises  must  be 
transposed  in  order.     Each  syllogism  converts  into  that  syllogism 
of  the  first  figure  which  has  the  same  initial  letter.     G  is  an 
addition  of  my  own,  presently  described  ;  it  must  be  left  out  when 


the  old  system  is  to  be  just  represented.  R,  N,  T,  have  no  signi- 
fication. The  following  are  the  names  put  together  in  memorial* 

*  The  best  attainable  exposition  of  logic  in  the  older  form,  with  modern 
criticism,  is  Mr.  Mansel's  edition  of  Aldrich's  compendium.  Should  a  reader 
of  this  work  desire  more  copious  specimens  of  old  discussion,  he  may  perhaps 
succeed  in  obtaining  Crackanthorpe's  Logica  Libri  quinque  (4to,  1622  and 
1677).  Sanderson's  Logic  is  highly  scholastic  in  character.  For  a  compen- 
dium of  mediaeval  logic,  ethics,  physics,  and  metaphysics,  I  have  never  found 
anything  combining  brevity  and  completeness  at  all  to  compare  with  the 
Precepta  Doctrince  Logicce,  Ethicce,  &c.  of  John  Stierius,  of  which  seven  or 
eight  editions  were  published  in  the  seventeenth  century  (from  1630  to  1689, 
or  thereabouts)  and  several  of  them  in  London.  There  is  a  large  system  of 
the  older  logic  in  the  lustitutiones  Logicce  of  Burgersdicius,  and  a  great 
quantity  of  the  metaphysical  discussion  connected  with  the  old  logic  in 
Brerewood  de  Predicabilibus  et  Predicamentis.  As  all  these  books  were  printed 
in  England,  there  is  more  chance  of  getting  them  than  the  foreign  logical 
works,  which  are  very  scarce  in  this  country.  For  more  than  usual  infor- 
mation on  parts  of  the  history  of  logical  quantity,  a  subject  now  exciting  much 
attention,  see  Mr.  Baynes's  New  Analytic,  in  which  will  be  found  much 
valuable  history  so  completely  forgotten  that  it  is  as  new  as  if  he  had 
invented  it  himself. 

109.  Barbara,  Cela^rent,  Darii,  Feri^oque  prioris: 
Cesare^r,  Camestres,  Festino<jr,  Baroko  secundae : 
Tertia  Darapt<jri,  Disa^mis,  Dati^rsi,  Felapton, 
Bokardo,  Ferison  habet.     Quarta  insuper  addit 
Bramanti^rp,  Came<mes,  Dimari^s,  Fe^sapo,  Fre^rsison. 

110.  I  leave  the  verification  of  what   has  been  said  as   an 
exercise.     As  an  example  of  reduction  into  the  first  figure  take 
the  syllogism  Camestres  from  the  second  figure. 



111.  The  letter  Gr  indicates  (§  103)  the  member  in  which, 
when  a  transitive  but  not  simply  convertible  copula  is  used,  the 
copula  is  to  be  the  converse  of  the  copula  employed  in  the  other 
members.  Thus  Cela^rent  shows  that  the  minor  premise  must 
have  the  converse  copula.  Suppose,  for  example,  that  the  copula? 
are  gives  to  (transitively  understood)  and  receives  from.  Then 
'  No  Y  gives  to  an  X ;  every  Z  receives  from  a  Y '  yield  e  No  Z 
gives  to  an  X'.  For  if  any  X  received  from  a  Z,  which  (second 

II.   Camestres      reduced  into       I.  Celarent. 
Every  Z  is  Y                ,    ,         No       Y  is  X     E 
No       XisY    (s)                     Every  Z  is  Y     A 


X  is  Z     (s) 

No       ZisX     E 


premise)  receives  from  a  Y,  that  X  would  receive  from  a  Y,  which 
contradicts  the  first  premise. 

112.  In  the  preceding  articles  I  have  considered  hardly  any- 
thing but  mere  assertion  or  denial  of  concomitance,  of  any  sort  or 
kind  whatsoever.      I  now  proceed  to  more  specially  subjective 
views  of  logic. 


113.  A  term  or  name  may  be  in  one  word  or  in  many.     It 
describes,  pictures,  represents,  but  does  not  assert  nor  deny.     Its 
object  must  exist,  whether  in  thought  only,  or  in  external  nature 
as  well :  and  everything  which  does  not  contradict  the  laws  of 
thought  may  be  the  object  of  a  term.     But  sometimes  the  thinker's 
universe  will  be  the  whole  universe  of  thought ;  sometimes  only 
the  objective  universe  of  external  reality ;  sometimes  only  a  part 
of  one  or  the  other. 

114.  Terms  are  used  in  four  different  senses.     Two  objective, 
directed  towards  the  external  object,  or  to  use  old  phrases,  of  first 
intention,  or  representing  first  notions.     Two  subjective,  directed 
towards  the  internal  mind,  of  second  intention,  or  representing 
second  notions. 

115.  In  objective  use  the  name  represents,  1.  The  individual 
object,  unconnected  with,  and  unaggregated  with,  any  other  object 
of  the  same  name  ;  2.  The  individual  quality,  forming  part  of,  and 
residing  in,  the  individual  object.     One  name  may,  at  different 
times,  represent  both :  thus  animal,  the  name  of  an  object,  is  the 
name  of  a  quality  of  man.     In  fact,  quality  is  but  object  con- 
sidered as  component  of  another  object.     The  quality  white,  a 
component  of  the  notion  of  an  ivory  ball,  is  itself  an  object  of 

116.  The  objective*  uses  of  names  have  been  considered  as 
the  bases  of  propositions  and  syllogisms,  in  the  preceding  part  of 
this  tract. 

*  The  ordinary  syllogism  of  the  logicians,  literally  taken  as  laid  down 
by  them,  is  objective,  of  first  intention,  arithmetical.  I  call  it  the  logician's 
abacus.  When  the  educated  man  rejects  its  use,  and  laughs  at  the  idea  of 
introducing  such  learned  logic  into  his  daily  life,  I  hold  his  refusal  to  be  in 
most  cases  right,  and  his  reason  to  be  entirely  wrong.  He  has,  and  his  peers 
always  have  had,  some  command  of  the  subjective  syllogism,  the  combination 
of  relations  to  which  I  shall  come.  He  has  no  more  occasion — in  most  cases 
— to  have  recourse  to  the  logical  abacus  for  his  reasoning,  than  to  the 
chequer- board,  or  arithmetical  abacus,  for  adding  up  his  bills.  I  hold  the 
combination  of  relations  to  be  the  actual  organ  of  reasoning  of  the  world  at 
large,  and,  as  such,  worthy  of  having  its  analysis  made  a  part  of  advanced 

38  CLASS  AND  ATTRIBUTE.  [116-121. 

education;   the  logician's  abacus  being  a  fit  and  desirable  occupation  for 

117.  In  subjective  use  the  name  represents,   1.   A  class,  a 
collection  of  individual  objects,  named  after  a  quality  which  is  in 
thought  as  being  in  each  one :    2.   An  attribute,  the  notion  of 
quality  as  it  exists  in  the  mind  to  be  given  to  a  class.     Attribute 
is   to   individual    quality   what    class    is    to    individual    object. 
Between  the  notion  of  a  class,  and  the  notion  of  an  object,  though 
the  name  be  the  same  in  both  cases,  there  is  this  distinction.     The 
class-name  belongs  to  a  number  of  objects :  the  object  belongs  to 
a  number  of  class-names ;  for  it  may  be  named  after  any  one  of 
its  qualities.     We  have  classes  aggregated  of  many  objects :  and 
objects  compounded  of  many  classes.     But  in  this  second  case  the 
object  is  said  to  have  many  qualities.     The  class  is  a  whole  of  one 
kind:  the  object  is  a  whole  of  another  kind.     This  distinction 
emerges  the  moment  a  name  begins  to  be  a  universal,  that  is, 
belonging  to  more  than  one  object. 

118.  Class  and  attribute  are  units  of  thought:    a   noun   of 
multitude  is  not  a  multitude  of  nouns.     When  we  are  fortunate 
enough  to  get  four  distinct  names,  we  readily  apprehend  all  these 
distinctions.     This  happens  in  the  case  of  our  own  species :  the 
objects  men,  all  having  the  quality  human,  give  to  the  thought 
the  class  mankind,  distinguished  from  other  classes  by  the  attri- 
bute humanity.     Should  any  reader  object  to  my  account  of  the 
four  uses  of  a  name,  he  can,  without  rejection  of  anything  else  in 
what  follows,  substitute  his  own  account  of  the  four  words  man, 
human,  mankind,  humanity. 

119.  With  grounds  of  classification,  and  reasons  for  nomen- 
clature, logic  has  nothing  to  do.      Any  number  of  individuals, 
whether  yet  unclassed,  or  included  in  one  class,  or  partly  in  one 
class  and  partly  in  another,  may  be  constituted  a  new  class,  in 
right  of  any  quality  seen  in  all,  by  which  an  attribute  is  affixed 
to  the  class  in  the  mind. 

120.  The  term  X,  or  Y,  or  Z  may  and  does  denote,  at  one 
time  or  another,  the  individual,  the  quality,  the  class,  or  the 
attribute.     Any  one  who  finds  the  distinction  useful  might  think 
of  the  individual  X,  the  quality  X-ic,  the  class  X-kind,  and  the 
attribute  X-ity. 

121.  Identical  terms  are  those  which  apply  to  precisely  the 
same  objects  of  thought,  neither  representing  more  than  the  other: 

121-124.]  umvEKSE.  —  CONTKARIES.  39 

so  that  identical  terms  are  different  names  of  the  same  class. 
Thus,  for  this  earth,  man  and  rational  animal  are  identical  terms. 
The  symbol  X||Y  will  be  used  (§  57)  to  represent  that  X  and  Y 
are  identical.  When  of  two  identical  terms  one,  the  known,  is 
used  to  explain  the  other,  the  unknown,  the  first  is  called  the 
definition  of  the  second. 

122.  The  whole  extent  of  matter  of  thought  under  consider- 
ation I  call  the  universe.     In  common  logic,  hitherto,  the  universe 
has  always  been  the  whole  universe  of  possible  thought. 

123.  Every  term  which  is  used  (§   16)  divides  the  universe 
into  two  classes;  one  within  the  term,  the  other  without.     These 
I  call  contraries  :  and  I  denote  the  contrary  class  of  X,  the  class 
not-X,  by  x.     When  the  universe  is  unlimited  contrary  names 
are  of  little  effective  use  ;  not-man,  a  class  containing  every  thing 
except  man,  whether  seen  or  thought,  is  almost  useless.     It  is 
otherwise  in  a  limited  universe,  in  which  contraries,  by  separate 
definiteness  of  meaning,  cease  to  be  mere  negations  each  of  the 
other,  and  even  acquire  separate*  and  positive  names.    Thus,  the 
universe  being  property  under  English  law,  real  and  personal  are 
contrary  classes.     Logic  has  nothing  to  do  with  the  difficulties  of 
allotment  which  take  place  near  the  boundary:  with  the  decision 
upon  those  personals,  for  example,  which,    as  the  lawyer  says, 
savour  of  the  realty.      The  lawyer  must  determine  the  classifi- 
cation, and   logic   investigates  the  laws  of  thought  which  then 
apply.      If  the   lawyer   choose  to  make   an  intermediate  class, 
between  real  and  personal,  then  real  and  personal  are  no  longer 
logical  contraries. 

That  X  and  Y  are  contrary  classes  is   denoted  (§  57)  by 

*  The  most  amusing  instance  which  ever  came  within  my  own  knowledge 
is  as  follows.  A  friend  of  mine,  in  the  days  of  the  Irish  Church  Bill,  used 
to  discuss  politics  with  his  butcher  :  one  day  he  alluded  to  the  possible  fate 
of  the  Establishment.  'Do  you  mean  do  away  with  the  church'?  asked  the 
butcher.  'Yes',  said  my  friend,  'that  is  what  they  say'.  'Why,  sir,  how 
can  that  be'?  was  the  answer  ;  'don't  you  see,  sir,  that  if  they  destroy  the 
church,  we  shall  all  have  to  be  dissenters'! 

124.  Terms  may  be  formed*  from  other  terms,  — 

1.  By  aggregation,  when  the  complex  term  stands  for  every- 
thing to  which  any  one  or  more  of  the  simple  terms  applies.    Thus 
animal  is  the  aggregate  of  (the  aggregants]  man  and  brute. 

2.  By  composition,  when  the  complex  term   stands  but  for 


everything  to  which  all  the  simple  terms  apply.     Thus  man  is 
compounded  of  (the  components)  animal  and  rational. 
3.  By  mixture  of  these  two  methods  of  formation. 

*  The  reader  must  carefully  remember  that  we  are  now  engaged  (§  4) 
especially  upon  the  esse  quod  habent  in  anima :  and,  if  not  accustomed  to 
middle  Latin,  he  must  remember  also  that  esse  is  made  a  substantive,  mode 
of  being.  Animal  cannot  be  divided  into  man  and  brute  except  in  a  mind. 
Logical  composition  must  be  distinguished  from  physical  or  metaphysical. 
Light  always  consisted  of  the  prismatic  components ;  but,  before  Newton,  it 
was  not  a  logical  quality  of  light  that  it  is  to  be  conceived  as  decomposible. 
Accordingly,  a  compound  of  qualities,  though  it  may  constitute  a  full  dis- 
tinctive definition  of  an  object  of  thought,  can  never  be  accepted  as  a  full 
description :  there  may  be  many  more.  The  logician  therefore  must,  in 
thinking  of  a  compound,  imitate  the  genial  Dean  Aldrich,  the  author  of  the 
Compendium  of  Logic  to  which  so  many  have  been  indebted,  in  the  structure 
of  his  fifth  reason  for  drinking. 

125.  The   aggregate   of  X,  Y,  Z   will   be   represented    by 
(X,  Y,  Z) :  the  term  compounded  of  X,  Y,  Z  will  be  represented 
by  (X-Y-Z)  or  (XYZ). 

126.  The  aggregate  name  belongs  to  each  of  the  aggregants: 
but  the  compound  name  does  not  belong  to  each  of  the  com- 
ponents, necessarily. 

127.  An  aggregate  is  not  impossible  if  either  of  its  aggregants 
be  impossible,  or  if  two  of  them  be  contradictory :  but  a  compound 
is  impossible  in  either  of  these  cases. 

128.  In  these  and  all  other  formulae,  care  must  be  taken  to 
remember  that  the  logical  phrase  implies  nothing:  the  phrases  of 
ordinary  conversation  frequently  imply,  in  addition  to  what  they 
express.     Thus  *  some  living  men  breathe '  and  '  every  man  is 
either  animal  or  mineral '  are    colloquially  false   by  what  they 
imply,  but  logically  true  because  the  logical  use  implies  nothing. 

129.  A  class  may  be  compounded  of  classes,  as  well  as  aggre- 
gated :  thus  the  class  marine  is  compounded  of  the  classes  soldier 
and  sailor.     An  attribute  may  be  aggregated  of  attributes,  as  well 
as  compounded:  thus  Adam  Smith's  attribute  productive  is  ag- 
gregated of  land-tilling,  manufacturing,  &c.     But  composition  of 
classes  and  aggregation  of  attributes  are  infrequent.     Any  name 
may  be  thought  of  either  as  a  class,  or  an  attribute  (or  character, 
as  it  is  often  called):  and  it  is  usual  and  convenient  to  think  of 
class  when  aggregation  is  in  question,  and  of  attribute  when  com- 
position is  in  question.     So  we  rather  say  the  marine  unites  the 
cfiaracters  of  the  soldier  and  the  sailor :  the  productive  classes  (as 

129-134.]  EXTENSION  AND  INTENSION.  41 

Adam  Smith  said)  consist  of  farmers,  manufacturers,  &c.  But 
all  this  for  convenience,  not  of  necessity :  and  the  power  of  un- 
learning usual  habits  must  be  acquired.  All  modes  of  thought 
should  be  considered:  the  usual,  because  they  are  usual;  the 
unusual,  that  they  may  become  usual. 

130.  The  words  of  aggregation  are  either,  or:  of  composition, 
both,  and.     Thus  (X,  Y)  is  either  X  or  Y  (or  both)  ;  (XY)  is  both 
X  and  Y. 

131.  The  more  classes  aggregated,  so  long  as  each  class  has 
something  not  contained  in  any  of  the  others,  the  greater  the 
extension*  of  the  aggregate  term.      The  more  attributes  com- 
pounded, so  long   as   each   attribute   has  some  component   not 
contained  in  the  others,  the  greater  the  intension.     Animal  has 
more    extension    than    man :     man    has    more    intension    than 

*  The  logicians  have  always  spoken  of  'all  men'  as  constituting  the 
'extent'  of  the  term  man  :  thus  the  whole  extent  of  man  is  part  of  the  extent 
of  animal.  They  have  chosen  that  their  more  and  less  should  be  referred  to 
by  phrases  derived  rather  from  the  notion  of  area  than  from  that  of  number. 
Hence  arise  certain  forms  of  speech  which,  when  quantity  is  applied  to  the 
predicate,  are  not  idiomatic;  as  'All  man  is  some  animal'. 

There  are  savage  tribes  which  have  not  sufficient  idea  of  number  even 
for  their  own  purposes :  among  them,  when  a  dozen  or  more  of  men  are  to  be 
indicated,  an  area  sufficient  to  contain  them  is  marked  out  on  the  ground. 
The  speculative  philosophers  of  the  middle  ages  were  in  something  like  the 
same  position :  though  the  mercantile  world  was  well  accustomed  to  large 
numbers,  the  philosophical  world,  excepting  only  some  of  the  mathematicians, 
was  very  awkward  at  high  numeration.  The  works  on  theoretical  arithmetic 
show  this  well.  I  have  been  straining  my  eye  over  the  twelve  books  of  the 
Arithmetical  Speculativa  of  Gaspar  Lax  of  Arragon  (Paris,  folio,  1515)  to 
detect,  if  I  could,  a  number  higher  than  a  hundred ;  and  I  have  found  only 
one,  the  date. 

132.  The  name  of  greatest  extension,  and  of  least  intension,  of 
which  we  speak,  is  the  universe. 

133.  The  contrary  of  an  aggregate  is  the  compound  of  the 
contraries  of  the  aggregants :  either  one  of  the  two  X,  Y,  or  both 
not-X  and  not-Y ;  either  (X,  Y)  or  (xy).     The  contrary  of  a 
compound  is  the  aggregate  of  the  contraries  of  the  components ; 
either  both  X  and  Y,  or  one  of  the  two,  not-X  and  not-Y ;  either 
(XY),  or  (x,  y). 

134.  The  following  are  exercises  on  complex  terms, — 

'  Both  or  neither '  and  f  one  or  the  other,  not  both ',  are  con- 
traries.    That  is  (XY,  xy)  and  (Xy,  Yx)  are  contraries.     Now 



the  contrary  of  the  first  is  (x,  y)-(X,  Y),  which  is  (xX,  xY,  yX,  yY), 
which  is  (xY,  yX)  since  xX,  y Y,  are  impossible. 

X||(A,B)C  gives  x||(ab,c) 
X||(A,B)(C,D)  -  x|j(ab,cd) 
X||AB,C  —  x||(a,b)c 

X||A,B(C,D)     gives  x||a(b,_cd) 
X||(A,BC)(D,EF)-  x||(ab,c),(de,f) 
X||(A,B,aC)  —  x||abc 

Deduce  these,  and  explain  the  last. 

135.  A  term  given  in  extension,  as  (A,  B,  C),  has  its  contrary 
given  in  intension,  (abc);  and  vice  versa.     Aggregates  or  com- 
ponents of  either  only  enable  us  to  deduce  components  or  aggre- 
gates of  the  other. 

136.  A  proposition  is  the  presentation,  for  assertion  or  denial, 
of  two  names  connected  by  a  relation :  as  ( X  in  the  relation  L  to 
Y.'     A  judgment  is  the  sentence  of  the  mind  upon  a  proposition: 
certainly  true,  more  or  less  probable,  certainly  false.    Propositions 
without  accompanying  judgment  hardly  occur:    so  that  propo- 
sition comes  to  mean,  by  abbreviation,  proposition  accompanied  by 

137.  The  distinction  between  certainty   and   probability  is 
usually  treated  apart  from  logic,  as  a  branch  of  mathematics.     A 
few  of  the  leading  results,  relative  to  authority  and  argument,  will 
be  afterwards  given. 

138.  The  purely  formal  proposition  with  judgment,  wholly 
void  of  matter,  is  seen  in  '  There  is  the  probability  a  that  X  is 
in  the  relation  L  to  Y'.     From  the  purely  formal  proposition  no 
inference   can  follow.      In  all  elementary  logic,  the  terms  are 
formal,  the  relation*  material,  and  the  judgment  absolute  assertion 
or  denial  (or,  as  a  mathematician  would   say,  the  probabilities 
considered  are  only  1  and  0). 

*  The  logician  calls  '  Every  man  is  animal'  a  material  instance  of  the 
formal  proposition  'Every  X  is  Y'.  He  will  admit  no  relation  to  be  formal 
except  what  can  be  expressed  by  the  word  is :  he  declares  all  other  relations 
material.  Thus  he  will  not  consider  '  X  equals  Y '  under  any  form  except 
'  X  is  an  equal  of  Y '.  He  has  a  right  to  confine  himself  to  any  part  he 
pleases :  buj  he  has  no  right,  except  the  right  of  fallacy,  to  call  that  part  the 

139.  Contrary  propositions  are  a  pair  of  which  one  must  be 
true  and  one  false :  as  '  he  did ',  '  he  did  not ';  or  as  '  Every  X  is 
Y',  *  Some  Xs  are  not  Ys '.     Contraries  contradict*  one  another; 
but  so  do  other  propositions.     Thus  e  All  men  are  strong '  and 
'  all  men  are  weak '  contradict  one  another  to  the  utmost :  the 
second  says  there  is  not  a  particle  of  truth  in  the  first.     But  the 
contrary  merely  says  there  is  more  or  less  falsehood :  to  '  all  men 

139-146.]  INFERENCE  AND  PROOF.  43 

are  strong '  the  contrary  is  *  There  are  [man  or]  men  who  are  not 

*  In  the  usual  nomenclature  of  logicians,  what  I  call  the  contrary  is 
called  the  contradictory,  as  if  it  were  the  only  one.     In  common  language, 
when  two  persons  disagree,  we  say  they  are  on  contrary  sides  of  the  question  : 
in  the  usual  technical  language  of  logic,  this  would  mean  that  if  one  should 
say  all  men  are  strong  the  other  says  no  man  is  strong.     But  in  common 
language,  the  one  who  maintains  the  contrary  is  he  who  advocates  anything 
which  the  other  is  opposed  to. 

140.  Every  proposition  has  its  contrary :  there  is  no  assertion 
but  has  its  denial;  no  denial  but  has  its  assertion.     Every  logical 
scheme  of  propositions  must  contain  a  denial  for  every  assertion, 
and  an  assertion  for  every  denial. 

141.  Inference  is  the  production  of  one  proposition  as  the 
necessary  consequence  of  one  or  more  other  propositions.     In- 
ference  from    one   proposition  may  be  either  an  equivalent  or 
identical  proposition,  or  an  inclusion.     If  from  a  first  proposition 
we  can  infer  a  second,  and  if  from  the  second  proposition  we  can 
also  infer  the  first,  the  two  propositions  are   logical  equivalents. 
Thus  '  X  is  the  parent  of  Y '  and  '  Y  is  the  child  of  X '  are  logical 
equivalents :    And  also  '  Every  X  is  Y '  and  '  Every  not-Y  is 
not-X '.     But  from  *  Every  X  is  Y  '  we  can  infer  f  Some  Xs  are 
Ys ',  without  being  able  to  infer  the  first  from  the  second :  the 
second  is  only  included  in  the  first. 

142.  When  inference  is  made  from  more  than  one  proposition, 
the  result  is  called  a  conclusion,  and  its  antecedents  premises. 

143.  Inference  has  nothing  to  do  with  the  truth  or  falsehood 
of  the  antecedents,  but  only  with  the  necessity  of  the  consequence. 
When  the  inference  from  the  antecedents  is  preceded  by  showing 
of  their  truth,  the  whole  is  called  proof  or  demonstration. 

144.  Deduction,  or  a  priori  proof,  is  when  the  compound  of  the 
premises  gives  the  conclusion.     One  false  premise,  and  deduction 
wholly  fails. 

145.  Induction,  or  a  posteriori  proof,  is  when  the  aggregate  of 
the  premises  gives  the  conclusion.     One  false  premise,  and  the 
induction  partially  fails. 

146.  Absolute  or  mathematical  proof  is  when  the  conclusion  is 
so  established  that  any  contradiction  would  be  a  contradiction  of 
a  necessity*  of  thought 

*  Logic  considers  the  laws  of  action  of  thoiight :    mathematics  applies 
these  laws  of  thought  to  necessary  matter  of  thought.     That  two  straight 
lines  cannot  inclose  a  space  is  a  necessary  way  of  thinking,  a  proposition  to 

44  LOGIC  AND  MATHEMATICS.  [146-147. 

which  we  must  assent :  but  it  is  not  a  law  of  action  of  thought.  That  if  two 
straight  lines  cannot  inclose  a  space,  it  follows  that  two  lines  which  do 
inclose  a  space  are  not  both  straight,  is  an  example  of  a  rule  by  which 
thought  in  action  must  be  guided. 

Mathematics  are  concerned  with  necessary  matter  of  thought.  Let  the 
mind  conceive  every  thing  annihilated  which  it  can  conceive  annihilated,  and 
there  will  remain  an  infinite  universe  of  space  lasting  through  an  eternity  of 
duration  :  and  space  and  time  are  the  fundamental  ideas  of  mathematics. 
Of  course  then  the  logicians,  the  students  of  the  necessary  action  of  thought, 
are  in  close  intellectual  amity  with  the  mathematicians,  the  students  of  the 
necessary  matter  of  thought.  It  may  be  so  :  but  if  so,  they  dissemble  their 
love  by  kicking  each  other  down  stairs.  In  very  great  part,  the  followers 
of  either  study  despise  the  other.  The  logicians  are  wise  above  mathematics; 
the  mathematicians  are  wise  above  logic :  of  course  with  casual  exceptions. 
Each  party  denies  to  the  other  the  power  of  being  useful  in  education :  at 
least  each  party  affirms  its  own  study  to  be  a  sufficient  substitute  for  the 
other.  Posterity  will  look  on  these  purblind  conclusions  with  the  smile  of 
the  educated  landholder  of  our  day,  when  he  reads  Squire  Western's  fears 
lest  the  sinking  fund  should  be  sent  to  Hanover  to  corrupt  the  English 
nation.  A  generation  will  arise  in  which  the  leaders  of  education  will  know 
the  value  of  logic,  the  value  of  mathematics,  the  value  of  logic  in  mathe- 
matics, and  the  value  of  mathematics  in  logic.  For  the  mind,  as  for  the 

body,  B/av   vrogi^ou  ircivTeS-i   srXjjv   \x  xaxut. 

This  antipathy  of  necessary  law  and  necessary  matter  is  modern.  Very 
many  of  the  most  illustrious  names  in  the  history  of  logic  are  the  names 
of  known  mathematicians,  especially  those  of  the  founders  of  systems,  and 
the  communicators  from  one  language  or  nation  to  another.  As  Aristotle, 
Plato,  Averroes  (by  report),  Boethius,  Albertus  Magnus  (by  report),  Ramus, 
Melancthon,  Hobbes,  Descartes,  Leibnitz,  Wolff,  Kant,  &c.  Locke  was  a 
Competent  mathematician  :  Bacon  was  deficient,  for  the  consequences  of 
which  see  a  review  of  the  recent  edition  of  his  work  in  the  Athenceum  for 
Sept.  11  and  18,  1858.  The  two  races  which  have  founded  the  mathematics, 
those  of  the  Sanscrit  and  Greek  languages,  have  been  the  two  which  have 
independently  formed  systems  of  logic. 

England  is  the  country  in  which  the  antipathy  has  developed  itself  in 
greatest  force.  Modern  Oxford  declared  against  mathematics  almost  to  this 
day,  and  even  now  affords  but  little  encouragement :  modern  Cambridge  to 
this  day  declares  against  logic.  These  learned  institutions  are  no  fools, 
whence  it  may  be  surmised  that  possibly  they  would  be  wiser  if  they  were 
brayed  in  a  mortar;  certainly,  if  both  were  placed  in  the  same  mortar,  and 
pounded  together. 

147.  Moral  proof  is  when  the  conclusion  is  so  established  that 
any  contradiction  would  be  of  that  high  degree  of  improbability 
which  we  never  look  to  see  upset  in  ordinary  life.  Among  the 
most  remarkable  of  moral  proofs  is  that  common  case  of  induction 
in  which  the  aggregants  are  innumerable,  and  the  conclusion 
being  proved  as  to  very  many,  without  a  single  failure,  the  mind 


feels  confident  that  all  the  unexamined  aggregants  are  as  true 
as  those  which  have  been  examined.  This  is  probable  induction : 
often  confounded  with  logical  induction. 

148.  A  proof  may  be  mixed :  it  may  be  deduction  of  which 
some  components  are  inductively  proved :  it  may  be  induction,  of 
which  some  aggregants  are  deductively  proved. 

149.  Failure  of  proof  is  not  proof  of  the  contrary. 

150.  If  any  number  of  premises  give  a  conclusion,  denial  of 
the  conclusion  is  denial  of  one  or  more  of  the  premises.    If  all  but 
one  of  the  premises  be  affirmed  and  the  conclusion  denied,  that 
one  premise  must  be  denied.     These  two  processes,  conclusion 
from  premises,  and  denial  of  one  premise  by  denying  the  con- 
clusion  and   affirming   all   the   other  premises,  may  be   called 

151.  Repugnant  alternatives  are  propositions    of  which   one 
must  be  true,  and  one  only.     If  there  be  two  sets  of  repugnant 
alternatives,  of  the  same  number  of  propositions  in  each,  and  if 
each  of  the  first  set  give  its  own  one  of  the  second  set  for  its 
necessary  consequence,  then  each  of  the  second  set  also  gives  its 
own  one  of  the  first  set  as  a  necessary  consequence.     Thus  if 
A,  B,  C,  be  repugnant  alternatives,  and  also  P,  Q,  R,  and  if  P  be 
the  necessary  consequence  of  A,  Q  of  B,  R  of  C,  then  A  is  the 
necessary  consequence  of  P,  B  of  Q,  C  of  R.      If  P  be  true, 
neither  B  nor  C  can  be  true ;  for  then  Q  or  R  would  be  true, 
which  cannot  be  with  P.     But  one  of  the  three  A,  B,  C,  must 
be  true:    therefore  A  is   true.      And   similarly  for  the   other 

152.  A  relation  is  a  mode  of  thinking  two  objects  of  thought 
together :  a  connexion  or  want  of  connexion.     Denial  of  relation 
is  another  relation:  and  the  two  are  contraries.     The  universe 
may  have  only  a  selection  from  all  possible  relations. 

153.  The  name  in  relation  is  the  subject:  the  name  to  which 
it  is  in  relation  is  the  predicate.     Thus  in  'mind  acting  upon 
matter '  mind  is  the  subject,  matter  the  predicate,  acting  upon  is  the 
relation.     When  the  relation  is  convertible,  subject  and  predicate 
are  distinguished  only  by  order  of  writing,  as  in  §  9. 

154.  All  judgments  (asserted  or  denied  relations)   may   be 
reduced  to  assertion  or  denial  of  concomitance  by  coupling  the 
predicate  and  the  relation  into  one  notion.    As  in  f  mind  is  a  thing 
acting  on  matter'  or  *  mind  is  not  a  thing  acting  on  matter'.     In 
all  works   of  logic,  the  consideration   of  relation  in  general   is 

46  RELATION.  [154-162: 

evaded  by  this  transformation,  and  the  developement  of  the  science 
is  thereby  altogether  prevented. 

155.  If  X  be  in  some  relation  to  Y,  Y  is  therefore  in  some 
other  relation  to  X.     Each  of  these  relations  is  the  converse  of  the 
other.      Converse  relations  are   of  identical  effect,  and   neither 
exists   without  the  other.     In  conversion  the  subject  and  pre- 
dicate are  transposed  and  usually  change  order  of  mention  :  as  in 

*  X  is  master  of  Y ;  Y  is  servant  of  X'. 

156.  When  a  relation  is  its  own  converse,  it  is  said  to  be 
simply  convertible.     As  in  'X  has  nothing  in  common  with  Y' 
and  '  Y  has  nothing  in  common  with  X ';  or  as  in  '  X  is  equal  to 
Y '  and  <  Y  is  equal  to  X '. 

157.  When  the  subject  of  one  relation  is  made  the  predicate 
of  another,  the  first  predicate  may  be  made  the  predicate  of  a 
combined  relation:  as  in  the  master*  of  (the- nephew-of-Y),  that 
is,  the-master-of-the-nephew  of  Y. 

*  The  most  familiar  relations  are  those  which  exist  between  one  human 
being  and  another  ;  of  which  the  relations  of  consanguinity  and  affinity  have 
almost  usurped  the  name  relation  to  themselves.  But  hardly  a  sentence  can 
be  written  without  expression  or  implication  of  other  relations. 

158.  A  combined  relation  may  have  a  separate  name,  or  it 
may  not.     Thus  brother  of  parent  has  its  own  name,  uncle :  but 
friend  of  parent  has  no  name  which  describes  nothing  else. 

159.  A  combined  relation  may  be  of  limited  meaning,  or  it 
may  not.     Thus  '  non-ancestor  of  a  descendant  of  Z '  has  a  limit- 
ation of  meaning  with  reference  to  Z ;  he  is  certainly  non-ancestor 
of  Z,     But  '  ancestor  of  a  descendant  of  Z  '  has  no  such  limitation: 
any  person  may  be  the  ancestor  of  a  descendant  of  any  other. 

160.  When  a  relation  combined  with  itself  reproduces  itself, 
let  it  be  called  transitive:    as  superior;    superior  of  superior  is 
superior,  the  same   sort  of  superiority  being  meant  throughout. 
A  transitive  relation  has  a  transitive  converse:  thus  inferior  of 
inferior  is  inferior. 

161.  Relations  are  conceivable  both  in  extension  and  in  in- 
tension (§   131),  both  as  aggregates  and  as  compounds.     Thus 

*  child  of  the  same  parents  with '  is  aggregate  of '  brother,  sister, 
self:    the  relation  of  whole  to  part  has  among  its  components 
1  greater '  and  *  of  same  substance  with '. 

162.  If  two  relations  combine*   into  what  is  contained  in 
a  third  relation,  then  the  converse  of  either  of  the  two  combined 
with  the  contrary  of  the  third,  in  the  same  order,  is  contained  in 

162-163.]  RELATION. IDENTITY.  47 

the  contrary  of  the  other  of  the  two.  Thus  the  following  three 
assertions  are  identically  the  same,  superior  and  inferior  being 
taken  as  contraries,  that  is,  absolute  equality  not  existing.  Let 
the  combination  be  *  master  of  parent'  and  the  third  relation 
( superior '. 

Every  master  of  a  parent  is  a  superior 
Every  servant  of  an  inferior  is  a  non-parent 
Every  inferior  of  a  child  is  a  non-master. 

From  either  of  these  the  other  two  follow.  This  may  be  gene- 
rally proved :  at  present  it  will  be  sufficient  to  deduce  one  of  the 
assertions  before  us  from  another.  Assume  the  second;  from  it 
follows  that  every  parent  is  not  any  servant  of  an  inferior,  and 
therefore,  if  servant  at  all,  only  servant  of  superior,  whence  master 
of  parent  must  be  superior. 

*  This  theorem  ought  to  be  called  theorem  K,  being  in  fact  the  theorem 
on  which  depends  the  process  (§  108)  indicated  by  the  letter  K  in  the  old 
memorial  verses. 

163.  The  relation  in  which  an  object  of  thought  stands 
to  itself,  is  called  identity ;  to  every  thing  else,  difference.  Every 
thing  is  itself :  nothing  is  anything  but  itself :  and  any  two  things 
being  thought  of,  they  are  either  the  same  or  different,  and  can  be 
nothing  except  one  or  the  other.  These  principles  enter  into  the 
distinction  between  truth  and  falsehood :  but  cannot  distinguish 
one  truth  from  another.  They  are  antecedent*  to  all  nomen- 
clature, and  to  all  decomposition. 

*  Many  acute  writers  affirm  that  syllogism  can  be  evolved  from,  and 
solely  depends  upon,  three  principles:    1.  Identity,  A  is  A;     2.  Difference, 
A  is  not  not-A  ;   thirdly,  excluded  middle,  Every  thing  either  A  or  not- A. 
Now  syllogism  certainly  demands  the  perception  of  convertibility,  '  A  is  B 
gives  B  is  A',  and  of  transitiveness,  '  A  is  B  and  B  is  C  gives  A  is  C'.     Are 
the  two  principles  deducible  from  the  three  ?     If  so,  either  by  syllogism  or 
without.     If  by  syllogism,  then  syllogism,  before  establishment  upon  the 
three  principles,  is  made  to  establish  itself,  which  of  course  is  not  valid. 
Consequently,  we  must  take  the  writers  of  whom  I  speak  to  hold  that 
convertibility  and   transitiveness   follow  from  the  principles   of  identity, 
difference,  and  excluded  middle,  without  petitio  principii.     When  any  one  of 
them  attempts  to  show  how,  I  shall  be  able  to  judge  of  the  process :  as  it  is, 
I  find  that  others  do  not  go  beyond  the  simple  assertion,  and  that  I  myself 
can  detect  the  petitio  principii  in  every  one  of  my  own  attempts.    Until 
better  taught,  I  must  believe  that  the  two  principles  of  identity  and  transi- 
tiveness are  not  capable  of  reduction  to  consequences  of  the  three,  and  must 
be  assumed  on  the  authority  of  consciousness. 

Should  I  be  wrong  here :  should  any  logician  succeed,  without  assuming 
syllogism,  in  deducing  the  syllogism  of  the  identifying  copula  'is'  from 


what  may  be  called  the  three  principles  of  identification,  I  shall  then  admit 
a  completely  established  specific  difference  between  the  ordinary  syllogism 
and  others  in  which  the  copula,  though  convertible  and  transitive,  is  not  the 
substantive  verb.  I  should  expect,  in  such  an  event,  to  deduce  the  transi- 
tiveness  and  convertibility  of  'equals'  from  'A  equals  A',  'A  does  not 
equal  not- A'  and  'every  thing  either  equals  A  or  not- A',  where  A  is  magni- 
tude only. 

164.  Identity  is  agreement  in  every  thing  and  difference  in 
nothing.      Complex  objects   of  thought  usually  agree  in   some 
things  and  differ  in  others.     They  get  the  same  names  in  right 
of  those  points  in  which  they  agree,  and  different  names  in  right 
of  those  points  in  which  they  differ.     And  thus,  all  resemblances 
or  agreements  giving  an  agreement  of  names,  and  all  differences 
giving  a  difference  of  names,  all  the  forms  of  inference  are  capable 
of  being  evolved  out  of  those  forms  in  which  nothing  but  con- 
comitance or  non-concomitance  of  names  is  considered  (§  5  to 

165.  Relations  which  have  immediate  reference  to,  or  are 
directly  evolved  from,  the  application  of  names  and  the  mode  of 
thinking  about  names  in  connexion  with  objects  named,  or  with 
other  names,  may  be  called  onymatic  *  relations. 

*  The  logician  has  hitherto  denied  entrance  to  every  relation  which  is 
not  onymatic ;  declaring  all  others  to  be  material,  not  formal.  When  the 
distinction  of  matter  and  form  is  so  clearly  defined  that  it  can  be  seen  why 
and  how  no  connexions  are  of  the  form  of  thought  except  those  which  I 
have  called  onymatic,  it  will  be  time  enough  to  attempt  a  defence  of  the 
introduction  of  other  relations.  In  the  meantime,  looking  at  all  that  is 
commonly  said  upon  the  distinction  of  form  and  matter,  I  am  strongly 
inclined  to  suspect  that  there  is  nothing  but  a  mere  confusion  of  terms ;  that 
is,  that  when  the  logician  speaks  of  the  distinction  of  form  and  matter,  he 
means  the  distinction  of  onymatic  and  non-onymatic.  Dr.  Thomson,  in  his 
Outlines,  Sfc.  (§  15,  note)  observes  that  the  philosophic  value  of  the  terms 
matter  and  form  is  greatly  reduced  by  the  confusion  which  seems  invariably 
to  follow  their  extensive  use.  The  truth  is  that  the  mathematician,  as  yet, 
is  the  only  consistent  handler  of  the  distinction,  about  which  nevertheless 
he  thinks  very  little.  The  distinction  of  form  and  matter  is  more  in  the 
theory  of  the  logician  than  in  his  practice :  more  in  the  practice  of  the 
mathematician  than  in  his  theory. 

166.  The  only  relation   in  which  a  name,  as  a  name,  can 
stand  to  an  object,  is  that  of  applicable  or  inapplicable. 

167.  Names  may  have  many  grammatical  and  etymological 
relations  to  one  another,  but  the  only  relations  which  are  of  any 
logical  import  are  the  relations  in  which  they  stand  to  one  another 
arising  out  of  the  relations  in  which  they  stand  to  objects.     Ac- 


cordingly  we  consider  two  names  as  having  objects  to  which  both 
apply,  or  as  both  applying  to  nothing  whatsoever. 

168.  When  X,  Y,  Z,  are  individual  names,  and  we  say  <X  is 
Y,  Y  is  Z,  therefore  X  is  Z ',  we  can  but  mean  that  in  speaking  of 
X  and  Y  we  are  speaking  of  one  object  of  thought,  and  the  same  of 
Y  and  Z,  so  that  in  speaking  of  X  and  Z  we  are  speaking  of  one 
object.     The  law  of  thought  which  acts  in  this  'inference  is  the 
transitiveness  (§  160)  of  the  notion  of  concomitancy :  if  X  go  with  Y, 
and  Y  go  with  Z,  then  X  goes  with  Z. 

169.  When  names  denote  classes,  the  primary  relation  between 
them  is  that  of  containing  and  contained,  in  the  sense  of  aggregate 
and  aggregant(§  124):  other  relations  spring  out  of  this,  as  will 
be  seen.     This  relation  is  mathematical  in  its  character :  a  class  is 
made  up  of  classes,  just  as  an  area  is  made  up  of  areas.     It  is 
physically  possible  to  connect   the   two    aggregations :    we  can 
imagine  all    men  on  one  area,  and  all   brutes  on   another:   the 
aggregate  of  the  areas  contains  the  class  animal,  the  aggregate  of 
man  and  brute. 

170.  When  names  denote  attributes,  the  primary  relation  is 
that  of  containing  and  contained,  in  the  sense  of  compound*  and 
component  (§  124).     This  relation  is  metaphysical  in  its  character : 
the  mode  of  junction  of  components  is  not  mathematical,  but  is  a 
subject  for  metaphysical  discussion,  though  how  that  discussion 
may  terminate  is  of  no  importance  for  logical  purposes.      The 
manner  in  which  the  sources  of  the  notion  rational  are  combined 
with  those  of  the  notion  animal  in  the  object  which  is  called  man 
has  nothing   to   do  with  the  laws  of  thought  under  which  the 
compound  and  the  components  are  and  must  be  treated. 

*  It  is  not  uncommon  among  logical  writers  to  declare  that  an  attribute 
is  the  sum  of  the  attributes  which  it  comprehends ;  that,  for  example,  man, 
completely  described  by  the  notions  animal  and  rational  conjoined,  is  the 
sum  of  those  notions.  This  is  quite  a  mistake  :  let  any  one  try  to  sum  up 
animal  and  rational  into  man,  in  the  obvious  sense  and  manner  in  which  he 
sums  up  man  and  brute  into  animal.  The  distinction  of  aggregation  and 
composition,  very  little  noticed  by  logicians,  if  at  all,  runs  through  all  cases 
of  thought.  In  mathematics,  it  is  seen  in  the  distinction  of  addition  and 
multiplication ;  in  chemistry,  in  the  distinction  of  mechanical  mixture  and 
chemical  combination ;  in  an  act  of  parliament,  in  the  distinction  between 
'And  be  it  further  enacted'  and  'Provided  always';  and  so  on. 

Hartley  has  more  nearly  than  any  other  writer  produced  the  notion  of 
composition  as  distinguished  from  aggregation.  His  compound  idea  has  a 
force  and  meaning  of  its  own,  which  prevents  our  seeing  the  components  in 
it,  just  as,  to  use  his  own  illustration,  the  smell  of  the  compound  medicine 



overpowers  the  smells  of  the  ingredients.    But  even  Hartley  represents  the 
compound  of  A  and  B  by  A  +  B. 

171.  When  the  class  X  is  contained  in  the  class  Y,  as  an 
aggregant,  the  attribute  Y  is  contained  in  the  attribute  X,  as  a 
component.     Thus  the  class  man  is  contained  in  the  class  animal : 
the  attribute  animal  is  contained  in  the  attribute  man.     These 
two  apparent  contradictions  are  both  true  in  their  different  senses : 
say  he  is  man,  you  say  he  is  in  animal ;  say  he  is  man,  you  say 
animal  is  in  him.     Class  man  is  in  class  animal,  as  aggregant 
in  aggregate ;  attribute  animal  is  in  attribute  man,  as  component 
in  compound. 

172.  In  all  things  which  do  not  depend  on  ourselves,  we  learn 
to  think  of  that  which  always  happens  as  necessarily  happening,  of 
that  which  always  accompanies  as  being   essential,*  part  of-  the 
essence,  part  of  the  being.     This  metaphysical  notion  is  always  in 
thought,  in  one  form  or  other,  whenever  undeviating  concomitance 
of  one  notion  with  another  is  established  or  supposed. 

*  Upon  this  word  may  be  said,  once  for  all,  what  is  to  be  said  concerning 
the  use  of  metaphysical  terms  in  logic.  We  have  nothing  to  do  with  the 
way  in  which  the  mind  comes  to  them ;  our  affair  is  with  the  way  in  which 
the  mind  works  from  them.  Thus  it  is  absolutely  essential  to  the  fitness  of 
three  straight  lines  to  be  the  sides  of  a  triangle  that  any  two  should  be 
together  greater  than  the  third ;  contradiction  is  inconceivable.  It  is 
naturally  essential  to  an  apple  to  be  round;  contradiction  is  unknown  in 
nature.  It  is  commercially  and  conveniently  essential  to  a  tea-pot  to  have 
a  handle ;  any  contradiction  would  be  unsaleable  and  unusable.  In  all 
these  cases,  and  whatever  may  be  the  force  of  the  word  essential,  the  mode  of 
inference  is  the  same :  for  the  logical  consequences  of  Y  being  an  essential  of 
X  are  but  those  of  Y  being  always  found  whenever  X  is  found.  Why  then 
do  we  not  confine  ourselves  to  this  last  notion,  leaving  the  character  of  the 
conjunction,  be  it  a  necessity  of  thought,  a  result  of  uncontradicted  observa- 
tion, or  a  conventional  arrangement,  &c.  entirely  out  of  view?  Simply 
because,  by  so  doing,  we  fail  to  make  logic  an  analysis  of  the  way  in  which 
men  actually  do  think.  If  men  will  be  metaphysicians — and  metaphysicians 
they  will  be — it  must  be  advisable  to  treat  the  metaphysical  views  of  the 
most  common  relations,  the  onymatic,  in  a  system  of  logic.  The  metaphysical 
notion  is  a  natural  growth  of  thought,  and  children  and  uneducated  persons 
are  more  strongly  addicted  to  it  than  educated  adults. 

173.  Out  of  these   onymatic   relations   arise  five   different 
modes  of  enunciating  the  same  proposition.     One  of  these,  the 
arithmetical,  already  treated,  merely  states,  or  sums  up,  an  enu- 
meration of  concomitances,  or  non-concomitances :  as  in  *  Every 
man  is  an  animal ' ;  or  as  in  *  No  man  is  a  vegetable '. 


174.  The  four  subjective  modes  of  speaking  which  the  notions 
of  relation  develope,  are, — 

1.  Mathematical.    Here  both  subject  and  predicate  are  notions 
of  class :   the  class  man  contained  in  the  class  animal. 

2.  Physical.     The  subject  a  class,  the  predicate  an  attribute. 
As  in  'man  is  mortal':  the  class  man  has  the  attribute  subject  to 

3.  Metaphysical.     Both  subject  and  predicate  are  notions  of 
attributes.    As  in  *  humanity  is  fallible':  fallibility  a  component  of 
the  notion  humanity. 

4.  Contraphysical.     The  subject  an  attribute,  the  predicate  a 
class.    As*  in  'All  mortality  in  the  class  man',  or  'none  but  men 
are  mortal ':  that  is,  we  must  attribute  mortality  only  in  the  class 
man ;  or,  all  of  which  mortality  is  the  attribute  is  in  the  class 

*  I  take  a  falsehood  for  once,  to  remind  the  reader  that  with  truth  or 
falsehood  of  matter  we  have  nothing  to  do. 

175.  All  these  niodes  of  reading  are  concomitant :  each  one  of 
the  five  gives  all  the  rest.     If  all  the  men  in  the  universe*  be  so 
many  animals,  then  the  class  man  is  in  the  class  animal,  and  has 
the  attribute  animal  as  one  of  its  class  marks ;  also,  the  attribute 
animal  is  an  essential  of  the  attribute  humanity,  and  the  attribute 
humanity  is  to  be  sought  only  in  the  class  animal. 

*  According  to  the  universe  understood,  so  is  the  mode  of  taking  the. 
meaning  of  the  ouymatic   terms.      For  example,  if  the  universe  be  the 
universe  of  objective  reality,  then,  all  existing  men  being  ascertained  to  be 
animals,  it  is  of  the  nature  of  man,  as  actually  created,  to  be  animal :  the 
attributes  of  animal  are  naturally  essential  to  man.    If  the  universe  be  the 
universe  of  all  possible  thought,  then,  if  all  men  conceivable  be  animals,  if, 
for  whatever  reason,  it  be  impossible  to  think  of  man  without  thinking  of 
animal,  then  the  attribute  animal  is  an  essential  of  the  attribute  humanity. 
And  now  arises  a  question  of  words,  with  which  logic  has  nothing  to  do. 
Those  creatures  of  thought  which  occur  in  the  fables,  dogs  and  oxen,  &c. 
which  are  rational  as  well  as  animal,  are  they  men  f   Certainly  not,  according 
to  the  notion  which  the  word  represents.     Consequently,  the  phrase  rational 
animal  is  a  larger  term  than  man,  when  all  the  possibilities  of  thought  are 
in  question.     But  this  is  not  a  question  of  logic.     The  logician,  as  such,  does 
not  know  what  man  is,  nor  what  animal  is  : '  but  he  knows  how  to  combine 
'every  man  is  animal'  with  other  propositions,  so  soon  as  he  knows  that  he 
is  permitted  to  use  that  proposition. 

176.  We  have  now  to  render  the  proposition  and  the  syllogism 
into  the  four  readings,  mathematical,  metaphysical,  or  mixed,  in- 


venting  appropriate  terms  for  all  the  relations  which  occur.  It 
will  be  sufficient,  however,  to  treat  the  first  and  third  system,  the 
wholly  mathematical,  and  the  wholly  metaphysical. 

177.  When  Every  X  is  Y,  X))Y  or  Y((X,  let  the  class  X  be 
called  a  species  of  the  class  Y,  and  Y  a  genus*  of  X.     In  the  con- 
trary case,  when  some  Xs  are  not  Ys,  X('(Y  or  Y)')X,  let  X  be 
an  exient  of  Y,  and  Y  a  deficient  of  X. 

*  In  the  common   use  of  these  words,  the  species  is  a  part  only  of 
the  genus.    As  here  used   the  species  may  be  the  whole  genus.     This  is, 
to  my  mind,  the  greatest  liberty  I  have  taken  with  the  ordinary  terms  of 

178.  When  No  X  is  Y,  X)-(Y  or  Y)-(X,  let  each  class  be 
called  an  external  of  the  other,  or  let  the  two  be  called  coexternals. 
In  the  contrary  case,  when  some  Xs  are  Ys,  X()Y  or  Y()X, 
let  each  be  called  a  partient  of  the  other,  or  let  the  two  be  called 

179.  When  every  thing  is  either  X  or  Y,  X(-)Y  or  Y(-)X, 
let  each  class  be  called  a  complement  of  the  other.    In  the  contrary 
case,  when  some  things  are  neither  Xs  nor  Ys,  X)(  Y  or  Y)(X,  let 
each  class  be  called*  a  co-inadequate  of  the  other. 

*  Punsters  are  respectfully  informed  that  the  reading  coin-adequate,  and 
all  jokes  legitimately  deducible  therefrom,  are  already  appropriated,  and  the 
right  of  translation  reserved. 

180.  The  spicular  symbols  may  be  made  to  stand  for  the 
relations  themselves.     Thus  ))  means  species  or  genus,  according 
as  it  is  read  forwards  or  backwards ;  (( ,  genus  or  species  :  and  so 

181.  Genus  and  species  are  converse  relations;   as  also  exient 
and  deficient :  of  external,  partient,  complement,  coinadequate,  each 
is  its  own  converse.     Genus  and  deficient  are  contrary  relations ; 
as  are  species  and  exient,  external  and  partient,  complement  and 

182.  Genus  is  both  partient  and  coinadequate ;  as  also  is  species. 
External  is  both  exient  and  deficient,  and  so  is  complement. 

183.  These  are  exercises  in  the  meanings  of  the  terms,  and 
should  be  thought  of  until  their  truth  is  familiar;    as  also  the 
following, — 

The  genus  has  the  utmost  partience,  and  may  have  the  utmost 
coinadequacy.  The  species  has  the  utmost  coinadequacy,  and  may 
have  the  utmost  partience.  The  external  has  the  utmost  deficiency, 


and  may  have  the  utmost  exience.    The  complement  has  the  utmost 
exience,  and  may  have  the  utmost  deficiency. 

184.  These  relations  have  terminal  ambiguity,  founded  on  the 
notion  of  contained  having  two  cases,  filling  the  whole,  or  filling 
only  a  part.     Thus 

Genus  is  either  -species  or  exient 
Species  is  either  genus  or  deficient 
External  is  either  complement  or  coinadequate 
Complement  is  either  external  or  partient. 

185.  Read  the  identities  in  §  25  into  this  language,  as   in, 
Species  is  external  of  contrary,  Contrary  of  species  is  complement, 
Contraries  of  species  and  genus  are  genus  and  species,  &c. 

186.  The  following  are  the  combinations*  of  mathematical 
relation  which  take  place  in  syllogisms.     Each  triad  in  the  first 
list  contains  a  universal  and  two  particular  syllogisms,  the  three 
being  opponents  (§  47),  connected  also  by  the  theorem  in  §  162. 
The  second  list  (§  187)  contains  the  strengthened  syllogisms. 

))    ))          Species  of  species  is  species 
((    (•  (         Genus  of  exient  is  exient 
(•  (  ((          Exient  of  genus  is  exient 

((    ((  Genus  of  genus  is  genus 

))    )*)         Species  of  deficient  is  deficient 

)•)  ))  Deficient  of  species  is  deficient 

)•(  (•)  External  of  complement  is  species 
)•(  (•  (  External  of  exient  is  coinadequate 
(•  (  (•)  Exient  of  complement  is  partient 

(•)   )•(          Complement  of  external  is  genus 
(•)   )•)         Complement  of  deficient  is  partient 
)•)  )•(          Deficient  of  external  is  coinadequate 

))    )•(          Species  of  external  is  external 
((    ( )          Genus  of  partient  is  partient 
()    )•(         Partient  of  external  is  exient 

((    (•)         Genus  of  complement  is  complement 
))    )(  Species  of  coinadequate  is  coinadequate 

)(    (•)         Coinadequate  of  complement  is  deficient 

External  of  genus  is  external 
External  of  partient  is  deficient 
Partient  of  species  is  partient 


(•)  ))          Complement  of  species  is  complement 
(•)   )(  Complement  of  coinadequate  is  exient 

)(     ((  Coinadequate  of  genus  is  coinadequate. 

*  Note  that  when,  and  only  when,  one  of  the  combining  words  is  either 
genus  or  species,  the  other  two  words  are  the  same ;  and  this  throughout 
the  fundamental  or  unstrengthened  syllogisms.  What  law  of  thought  does 
this  represent  ?  And  except  when  one  of  these  words  so  occurs,  the  three 
words  of  relation  are  all  different. 

187.  ((    ))          Genus  of  species  is  partient 

))    ((  Species  of  genus  is  coinadequate 

(•)  (•)  Complement  of  complement  is  partient 

)•(  )•(  External  of  external  is  coinadequate 

((    )•(  Genus  of  external  is  exient 

))    (•)  Species  of  complement  is  deficient 

(•)  ((  Complement  of  genus  is  exient 

)•(  ))  External  of  species  is  deficient. 

188.  When  we  give  what   may  be   called,   comparatively, 
terminal  precision,  as  in  §  57,  we  may  use  the  following  nomen- 

.    )°)     A  deficient  species  may  be  called  a  subidentical 
1 1       A  species  and  genus  is  an  identical 
(o(     An  exient  genus  may  be  called  a  superidentical 

)o(     A  coinadequate  external  may  be  called  a  subcontrary 
I*  |       An  external  complement  is  a  contrary 
(°)     A  partient  complement  may  be  called  a  supercontrary. 

189.  The  complex  syllogisms  (§61)  may  be  read  as  follows, — 
)o)  )o)     A  subidentical  of  a  subidentical  is  a  subidentical 

(°(  (°(  A  superidentical  of  a  superidentical  is  a  superidentical 

)°(   (°)  A  subcontrary  of  a  supercontrary  is  a  subidentical 

(o)  )o(  A  supercontrary  of  a  subcontrary  is  a  superidentical 

)o)  )o(  A  subidentical  of  a  subcontrary  is  a  subcontrary 

(°(  (o)  A  superidentical  of  a  supercontrary  is  a  supercontrary 

)o(  (o(  A  subcontrary  of  a  superidentical  is  a  subcontrary 

(o)  )o)  A  supercontrary  of  a  subidentical  is  a  supercontrary. 

The  following  modes  of  connecting  the  symbols,  as  applied  to 
the  same  two  terms,  may  be  useful, — 
)°)       ))>  Species;  )•),  but  not  the  greatest  possible. 

(°(        ((,  Genus;  (•(,  but  not  the  least  possible. 

)o(       )•( ,  External ;          )(  ,  but  not  the  greatest  possible, 
(o)       (•),  Complement;   (),  but  not  the  least  possible. 


1 90.  I  now  proceed  to  the  metaphysical  relations  *  between 
attribute  and  attribute. 

*  The  terms  of  metaphysical  relation  are  picked  up  without  difficulty 
in  our  common  language  :  but  those  of  mathematical  relation  had  in  several 
instances  to  be  forged.  This  means  that  the  world  at  large  has  more  of  the 
metaphysical  than  of  the  mathematical  notion  in  its  usual  form  of  thought. 
But  though  the  unconnected  words  essential,  dependent,  repugnant,  alternative, 
are  constantly  on  the  tongues  of  educated  people,  the  combinations  of  these 
relations  are  not  made  with  any  security,  and  when  thought  of  at  all,  enter 
under  a  cloud  of  words :  while  the  analysis  by  which  precision  of  speech  and 
habit  of  security  might  be  gained  is  treated  with  contempt,  as  being  logic. 
A  whole  drawing-room  of  educated  men  may  be  without  a  single  person 
who  can  expose  the  falsehood  of  the  assertion  that  the  essential  of  an 
incompatible  must  be  incompatible ;  a  proposition  which  I  have  heard 
maintained,  though  not  in  those  words,  by  persons  of  more  than  respectable 
acquirements ;  sometimes  by  actual  error,  sometimes  by  confusion  between 
the  essential  of  an  incompatible,  and  that  to  which  an  incompatible  is 
essential.  But  even  of  the  persons  who  are  not  thus  taken  in,  very  few 
indeed,  when  told  that  the  answer  to  '  the  essential  of  an  incompatible  is 
incompatible'  is  'not  so  much;  only  independent',  will  be  puzzled  by  the 
juxtaposition  of  incompatibility  and  independence  as  viewed  in  a  relation  of 
degree.  In  making  these  remarks,  it  will  be  remembered  that  I  am  not 
speaking  of  any  words  of  my  own,  nor  of  any  meanings  of  my  own.  The 
words  are  common,  and.  I  take  them  in  their  common  meanings ;  but  it  is 
not  generally  seen  that  these  common  words,  used  in  their  common  senses, 
are  sufficient,  in  conjunction  with  their  contraries,  to  express  all  the 
relations  which  occur  in  a  completely  quantified  system  of  onymatic 

191.  "When  X))Y  let  the  attribute  Y  be  called  an  essential 
of  the  attribute  X,  and  X  a  dependent  of  Y.     In  the  contrary  case, 
X(-  ( Y,  let  Y  be  called  an  inessential  of  X,  and  X  an  independent 
of  Y.     Remember  that  dependent  on  does   not  mean  dependent 
wholly  on,  or  dependent  only  on. 

192.  When  X)-(Y,  let  each  attribute  be  called  a  repugnant  of 
the  other.     When  X(  )Y,  let  each  be  an  irrepugnant  of  the  other. 

193.  When  X(-)Y  let  each  attribute  be  called  an  alternative 
of  the  other.    When  X)(  Y  let  each  be  called  an  inalternative  of  the 

194.  When  difference  of  symbols  is  desired,  the  square  bracket 
may  be  used  instead  of  the  parenthesis :    thus  ]  ]  may  denote 
dependent  when   read  forwards,  and  essential  when  read  back- 
wards, &c. 

195.  Essential  and  dependent  are  converse  relations;  as  are 
also  inessential  and  independent.     Of  repugnant,  irrepugnant,  alter- 

56  METAPHYSICAL  SYLLOGISM.  [  1 95-200. 

native,  inalternative,  each  is  its  own  converse.  Essential  and 
inessential  are  contrary  relations ;  as  are  dependent  and  inde- 
pendent, repugnant  and  irrepugnant,  alternative  and  inalter- 
native. Compare  §  181. 

196.  The  essential  is  both  irrepugnant  and  inalternative:    as 
also  is  the  dependent.      The  repugnant  is  both  independent  and 
inessential:  as  also  is  the  alternative.     Compare  §  182. 

197.  The  essential  has  the  utmost  irrepugnance,   and   may 
have  the   utmost    inalternativeness.      The    dependent    has    the 
utmost  inalternativeness,  and  may  have  the  utmost  irrepugnance. 
The  repugnant  has  the  utmost  inessential  ity,  and  may  have  the 
utmost  independence.      The   alternative   has   the   utmost  inde- 
pendence, and  may  have   the   utmost   inessentiality.      Compare 

198.  These  relations   also  have  terminal  ambiguity  (Com- 
pare §  184). 

Essential  is  either  dependent  or  independent 
Dependent  is  either  essential  or  inessential 
Repugnant  is  either  alternative  or  inalternative 
Alternative  is  either  repugnant  or  irrepugnant. 

199.  Read  the   identities  in  §  25  into  this  language,  as   in 
Dependent  is  repugnant  of  contrary,  contrary  of  dependent  is 
alternative,  contraries  of  dependent  and  essential  are  essential  and 
dependent,  &c. 

200.  The  following  are  the  combinations*  in  syllogism,  ar- 
ranged as  in  §  186. 

))    ))        Dependent  of  dependent  is  dependent 
((    (•  (       Essential  of  independent  is  independent 
(•  (  ((         Independent  of  essential  is  independent 

((    ((        Essential  of  essential  is  essential 

))    )•)       Dependent  of  inessential  is  inessential 

)•)  ))        Inessential  of  dependent  is  inessential 

)•(  (•)       Repugnant  of  alternative  is  dependent 
)•(   (•  (       Repugnant  of  independent  is  inalternative 
(•  (  (•)       Independent  of  alternative  is  irrepugnant 

(•)  )•(       Alternative  of  repugnant  is  essential 
(•)    )•)       Alternative  of  inessential  is  irrepugnant 
)•)  )•(        Inessential  of  repugnant  is  inalternative 


))  )•(  Dependent  of  repugnant  is  repugnant 

((  (  )  Essential  of  irrepugnant  is  irrepugnant 

( )  )•(  Irrepugnant  of  repugnant  is  independent 

((  (•)  Essential  of  alternative  is  alternative 

))  )(  Dependent  of  inalternative  is  inalternative 

)(  (•)  Inalternative  of  alternative  is  inessential 

Repugnant  of  essential  is  repugnant 
Repugnant  of  irrepugnant  is  inessential 
Irrepugnant  of  dependent  is  irrepugnant 

Alternative  of  dependent  is  alternative 
Alternative  of  inalternative  is  independent 
Inalternative  of  essential  is  inalternative. 

*  Note  that  when,  and  only  when,  one  of  the  combining  words  is  either 
essential  or  dependent,  the  other  two  words  are  the  same ;  and  this  throughout 
the  fundamental  or  unstrengthened  syllogisms.     What  law  of  thought  does 
this  represent  ?     And  except  when  one  of  these  words  so  occurs,  the  three 
words  of  relation  are  all  different. 

201.  ((  ))  Essential  of  dependent  is  irrepugnant 
))  ((  Dependent  of  essential  is  inalternative 
(•)  (•)  Alternative  of  alternative  is  irrepugnant 
)•(  )•(  Repugnant  of  repugnant  is  inalternative 
((  )•(  Essential  of  repugnant  is  independent 

))  (•)  Dependent  of  alternative  is  inessential 
(•)  ((  Alternative  of  essential  is  independent 
)•(  ))  Repugnant  of  dependent  is  inessential. 

202.  I  now  proceed  to  form  metaphysical*  terms  expressing 
relations  of  terminal  precision  (compare  §  188).     Let  an  inherent 
be  an  attribute  asserted;  let  an  excludent  be  an  attribute  denied; 
let  an  accident,  which  is  also  non-accident,  be  an  attribute  affirmed 
of  part  and  denied  of  the  rest.     Thus  of  man,  life  is  an  inherent, 
vegetation  an  excludent,  wisdom  an  accident  and  a  non-accident. 

*  This  new  formation  cannot  be  overlooked,  since  it  is  the  extension  of 
the  Aristotelian  system  of  predicables,  genus  and  species  (used  in  the  old 
sense)  and  accident,  to  the  system  in  which  contrary  terms  are  permitted. 
Otherwise,  the  relations  of  terminal  ambiguity,  compounded,  might  serve 
the  purpose. 

203.  Each  of  these  relations  may  be  either  generic  or  specific. 
Either  is  generic  when  it  applies  in  as  large  or  a  larger  degree  to 
a  larger  genus :  specific,  when  it  does  not  so  apply  to  any  larger 

58  PREDICABLES.  [203-205. 

genus.     This  being  premised,  the  following  relations  will  be  found 
correctly  stated, — 

>.  v       T  ,.  i  j         i  .      f  Specific  accident 

)o)       Inessential  dependent      is   •<     *,     J . 

{^Generic  non-accident 

Dependent  essential         is  Specific  inherent 

(o(       Independent  essential      is  Generic  inherent 

)o(       Inalternative  repugnant  is  Generic  excludent 

|-|        Repugnant  alternative     is  Specific  excludent 

,  N  i,         ,.  f  Generic  accident 

(o )       Irrepugnant  alternative  is   <   _      . . 

(.specific  non-accident. 

204.  The  following  are  examples  of  each  of  these  terms,  the 
universe  being  terrestrial  animal, — 

Specific  accident ;  generic  non-accident.  Lawyer  is  in  this  rela- 
tion to  man:  accident  and  non-accident,  because  an  attribute  of 
some  men,  and  not  of  others ;  specific  accident,  because  not  found 
in  the  additional  extent  of  any  genus  larger  than  man ;  generic 
non-accident,  for  the  same  reason. 

Specific  inherent.  Rational  is  in  this  relation  to  man:  inhe- 
rent, because  an  attribute  of  all ;  specific,  because  no  attribute  of 
the  additional  extent  in  a  larger  genus. 

Generic  inherent  Biped  is  in  this  relation  to  man;  inherent, 
because  an  attribute  of  all ;  generic,  because  an  attribute  of  the 
additional  extent  of  a  larger  genus. 

Generic  excludent.  Oviparous  is  in  this  relation  to  man;  ex- 
cludent, because  an  attribute  to  be  denied  of  man ;  generic,  because 
to  be  also  denied  of  the  additional  extent  of  some  larger  genera. 

Specific  excludent.  Dumb  (wanting  articulate  language  with 
meaning)  is  in  this  relation  to  man;  excludent,  because  to  be 
denied  of  man ;  specific,  because  not  to  be  denied  of  the  additional 
extent  of  any  larger  genus. 

Generic  accident;  specific  non-accident.  Naked  (not  artificially 
clothed)  is  in  this  relation  to  man;  accident  and  non-accident, 
because  some  are  and  some  are  not;  generic  accident,  because  an 
accident  of  the  additional  extent  of  larger  genera;  specific  non- 
accident,  because  not  non-accident  of  any  such  additional  extent. 

205.  When  either  of  the  relations  belongs  equally  to  a  term 
and  its  contrary,  it  may  be  called  universal.     Thus  an  attribute  of 
both  term  and  contrary  is  a  universal  inherent;  an  accident  and 
non-accident  of  both  term  and  contrary  is  a  universal  accident  and 
non-accident;  an  excludent  of  both  term  and  contrary  is  a  uni- 
versal excludent.     But  the  first  and  third  of  these  terms  are  chiefly 



of  use  in  defining  the  universe  :  the  second  is  that  relation  which 
we  suppose  until  some  contradiction  is  affirmed. 

206.  With  the  arithmetical  reading  in  extension  may  be  joined 
that  in  intension,  §  115,  131.  In  extension,  the  unit  of  enume- 
ration is  one  of  the  objects  all  of  which  aggregate  into  the  class : 
in  intension,  the  unit  of  enumeration  is  one  of  the  qualities  all  of 
which  compose  the  object.  The  following  is  the  system  of  arith- 
metical reading  in  intension:  naturally  connected  (§  129)  with 
the  metaphysical  mode  of  viewing  objects  of  thought.  The 
inversion  of  the  quantities,  presently  further  described,  will  be 
easily  seen ;  namely,  that  )X  and  X(  now  indicate  that  X  is  taken 
completely, *in  all  its  qualities;  while  X)  and  (X  indicate  that  X 
is  taken  incompletely,  in  some  (some  or  all,  not  known  which) 
of  its  qualities.  The  term  any  (§  22)  is  here  introduced  when 
grammatically  desirable. 

Arithmetical  reading  in  intension 
All  qualities  of  Y  are  some  qualities  of  X 
Some  qualities  of  Y  not  any  qualities  of  X 

All  things  want  either  some  qualities  of  X, 
or  some  qualities  of  Y 

Some  things  want  neither  any  quality  of  X, 
nor  any  quality  of  Y 

All  things  have  either  all  the  qualities  of  X 
or  all  the  qualities  of  Y 

Some  things  want  either  some  of  the  quali- 
ties of  X,  or  some  of  the  qualities  of  Y 

Some  qualities  of  Y  are  all  the  qualities 

of  X 
Any  qualities  of  Y  are  not  some  qualities 

of  X. 

Symbol    Metaphysical  reading 
X))Y   ,  X  dependent  of  Y 
X  independent  of  Y 

X  repugnant  of  Y 


XQY      X  irrepugnant  of  Y 



X  alternative  of  Y 

X  inalternative  of  Y 

X  essential  of  Y 

X  inessential  of  Y 

207.  I  now  proceed  to  further  consideration  of  the  subject  of 
quantity.     No  new  results  can  appear,  but  it  will  be  necessary 
both  to  adapt  the  old  results  to  the  more  subjective  view  of  logical 
process,  and  also  to  consider   the   distinctions  of  quantity  from 
new  points  of  view. 

208.  The  distinction  of  the  two  tensions,  extension  and  inten- 
sion (§  131),  or,  for  brevity,  extent  and  intent,  may  for  clearness 
(§  129)  be  applied  only  to  classes  and  attributes.      The  extent 
of  a  class  embraces  all  the  classes  of  which  it  is  aggregated :  the 
intent  of  an  attribute  embraces  all  the  attributes  of  which  it  is 

209.  A  class  may  be  subdivided  down  to  the  distinct  and  non- 

60  QUANTITY  OF  EXTENT  AND  INTENT.  [209-211. 

interfering  individual  objects  of  thought  of  which  it  is  composed : 
and  here  subdivision  must  stop.  But  it  is  not  for  human  reason 
to  say  what  are  the  simple  attributes  into  which  an  attribute  may 
be  decomposed:  the  decomposition  of  the  notion  rational,  for 
example,  into  distinct  and  non-interfering  component  notions,  is 
the  subject  of  an  old  controversy  which  will  perhaps  never  be 
settled.  But  this  difficulty  is  of  no  logical  importance. 

210.  The  relation  of  quantity  as  exhibited  in  the  arithmetical 
view  of  the  proposition  (§  13, 14),  giving  the  distinction  of  univer- 
sal and  particular  quantity,  as  it  is  commonly  expressed,  or  of 
total  and  partial  quantity,  as  I  have  expressed  it,  may  be  in  this 
part  of  the  subject  most  conveniently  attached  to  other  names. 
Let  the  terms  full  extent*  and  vague  extent  be  used  to  replace 
total  extension  and  partial  extension :  and  let  full  intent  and  vague 
intent  replace  total  intension  and  partial  intension. 

*  These  terms  are  convenient  from  their  brevity :  full  extent  is  shorter 
than  universal  extension.     But  they  are  still  more  useful  as  avoiding  the 
ambiguity  of  the  words  some,  particular,  partial,  which,  as  we  have  seen 
(§  14,  note  f)  misleads  even  the  highest  writers.      The  logical  opposition 
of  quantity  is  not  quantity  universal  and  quantity  not  universal,  but  quantity 
asserted  to  be  universal  and  quantity  not  asserted  to  be  universal.     Two  words 
cannot  be  found  which  express  the  opposition  of  undertaking  to  assert  and 
not  undertaking  to  assert  universality.     We  may  therefore  be  content  with 
full  and  vague,  which,  if  they  do  not  express  opposition,  at  least  do  not,  like 
universal  and  particular,  express  the  wrong  opposition. 

211.  Additional  extent  can  only  be  gained  by  a  new  aggregate 
containing  extent  which  is  not  in  the   collective  extent  of  the 
others  :  additional  intent  only  by  a  new  component  which  is  not 
in  the  joint  intent  of  the  others.     Thus  the  extent  of  the  class 
animal  is  not  augmented  by  the  aggregation  of  the  class  having 
volition,  if  the  universe  be  the  visible  earth.     Again,  the  intent* 
of  the  notion  plane  triangle  is  not  augmented  by  the  junction  of 
the  notion  capable   of  inscription  in  a  circle.      The   distinction 
between  these  real  and  apparent  augmentations  is  of  the  matter, 
not  of  the  form:    and  is  of  no  logical  import  except  this,  that 
when  we  say  that  a  new  aggregant  increases  extent,  and  a  new 
component   increases   intent,   we  must   be   prepared,    with    the 
mathematicians,  to  reckon  0  among  the  cases  of  quantity. 

*  There  is  a  remarkable  difference  between  extent  and  intent,  which, 
though  logically  nothing  at  all,  is  psychologically  very  striking.     Say  we 
discover  extent  hitherto  unknown,  without  the  necessity  of  reducing  intent 
to  include  it  within  a  class  thought  of.     Columbus  did  this  when  he  first 
was  able  to  add  the  class  American  to  the  classes  then  known  under  man. 


Here  is  nothing  beyond  what  was  possible  in  previous  thought,  which  could 
people  the  seas  to  any  extent.  But  when  we  add  intent  without  diminishing 
extent,  which  knowledge  is  doing  every  day,  we  cannot  conceive  beforehand 
what  kind  of  additions  we  shall  make.  A  beginner  in  geometry  gradually 
adds  to  the  intent  of  triangle,  which  at  first  is  only  rectilinear  three-sided 
figure,  the  components  —  can  be  circumscribed  by  a  circle — has  bisectors  of 
sides  meeting  in  a  point — has  sum  of  angles  equal  to  two  right  angles  — 
and  other  properties,  by  the  score.  The  distinction  is  that  class  aggregation 
joins  similars* — but  that  composition  of  attributes  joins  things  perfectly 
distinct,  of  which  no  one  can  predicate  anything  merely  by  what  he  knows 
of  another  thing.  When  the  old  logicians  threw  the  notion  of  intent  out 
of  logic  into  metaphysics,  they  were  guided  by  the  material  differences  of 
qualities,  and  did  not  apprehend  their  similarity  of  properties  a*  qualities. 

212.  The  distinction  of  extent  and  intent  has  found  its  way 
into  common  language,  in  the  words  scope*  and  force,  which  I 
shall  sometimes  use.     Thus  in  *  every  man  is  animal'  the  term 
man  is  used  in  all  its  scope,  but  not  in  all  its  force ;  a  person 
incognisant  of  some  of  the  components  of  the  notion  man,  that  is, 
of  the  whole  force  of  the  term,  might  have  the  means  of  knowing 
this  proposition.     But  animal  is  used  in  all  its  force,  and  not  in 
all  its  scope.     This  answers  to  saying  that  in  'Every  X  is  Y', 
the  term  X  is  of  full  extent  and  vague  intent ;  the  term  Y  is  of 
full  intent  and  vague  extent 

*  The  logicians,  until  our  own  day,  have  considered  the  extent  of  a  term 
as  the  only  object  of  logic,  under  the  name  of  the  logical  whole  :  the  intent 
was  called  by  them  the  metaphysical  whole,  and  was  excluded  from  logic. 
In  our  own  time  the  English  logical  writers,  and  Sir  William  Hamilton 
among  the  foremost,  have  contended  for  the  introduction  of  the  distinction 
into  logic,  under  the  names  of  extension  and  comprehension :  Hamilton  uses 
breadth  and  depth.  Now  I  say  that  in  the  perception  of  the  distinction 
between  scope  and  force,  as  well  as  in  other  things,  the  world,  which  always 
runs  after  quack  preparations,  has  ventured  for  itself  out  of  the  logical 
pharmacopoeia.  This  certainly  in  a  rude  and  imperfect  way :  and  without 
apprehension  of  any  theorems.  I  have  not  found,  though  I  have  looked  for 
it,  any  such  amount  of  recognition  that  the  greater  the  scope  the  less  the 
force  as  I  could  present  without  suspicion  of  the  aut  inveniam  out  faciam 
bias.  But  I  think  it  likely  enough  that  some  of  my  readers  may  casually 
pick  up  passages  which  show  a.  feeling  of  this  theorem. 

213.  The  quantity  considered  in  the  arithmetical  view  of  logic 
(§  5-111)  was  entirely  quantity  of  extent.     I  now  proceed  to  the 
comparison  of  extent  and  intent. 

214.  In  every  use  of  a  term,  one  of  the  tensions*  is  full,  and 
the  other  vague:  the  full  extent  and  the  full  intent  cannot  be 
used  at  one  and  the  same  time ;  and  the  same  of  the  vague  extent 
and  the  vague  intent.     Thus  X)  and  (X  must  stand  for  X  used 


in  full  extent  and  vague  intent :  and  )X  and  X(  for  vague  extent 
and  full  intent. 

The  proof  of  this  proposition  is  as  follows.  When  a  term  is 
full  in  extent,  we  can  abandon  or  dismiss  any  aggregant  of  that 
extent  we  please :  the  proposition,  though  reduced  or  crippled  by 
the  dismissal,  is  true  of  what  is  left :  but  we  may  not  annex  an 
aggregant  at  pleasure.  When  a  term  is  vague  in  extent,  we 
cannot  dismiss  any  aggregant  whatever:  for  we  know  not  by 
what  aggregant  the  proposition  is  made  true :  but  we  may  annex 
any  aggregant  at  pleasure :  for  we  do  not  thereby  throw  out  what 
makes  the  proposition  true,  even  if  we  annex  no  additional  truth ; 
and  we  do  not,  when  speaking  vaguely,  affirm  or  deny  of  any  one 
selected  aggregant.  And  as  the  extent  must  be  full  or  vague, 
and  we  must  be  either  competent  or  incompetent  to  dismiss  an 
aggregant  taken  at  pleasure,  and  must  be  either  competent  or 
incompetent  to  annex  one,  the  converses  follow  (§  151),  namely, 
that  when  we  are  competent  to  dismiss,  the  extent  is  full,  and 
when  we  are  incompetent  the  extent  is  vague :  and  also  that  when 
we  are  competent  to  annex,  the  extent  is  vague,  and  when  not, 
the  extent  is  full.  Precisely  the  same  proposition  may  be 
established  upon  the  intent  of  a  term,  and  its  components. 

Now  let  a  term  be  of  full  extent.  In  diminishing  the  extent, 
which  we  may  do,  we  can  so  do  it  as  to  augment  the  intent :  and 
if  we  be  competent  to  augment  the  intent,  that  intent  must  be 
vague,  as  just  proved.  Similarly,  if  a  term  be  of  vague  extent,  we 
are  competent  to  annex  an  aggregant,  that  is,  to  diminish  the 
intent ;  whence  the  intent  must  be  full.  And  the  same  may  be 
proved  in  like  manner  when  either  kind  of  intent  is  first  supposed 
instead  of  extent;  though  by  use  of  §  151  this  case  may  be  seen 
to  be  contained  in  that  already  treated.  And  the  learner  may 
gather  the  whole  from  instances.  Thus  A,  B))PQ  gives  A))PQ, 
andA))P;  but  not  A,B,C))PQ,  nor  A,B))PQR.  But  PQ))A,B 
does  not  give  P)))A,  B,  nor  PQ))A,  though  it  does  give  PQR))A,B 
and  PQ))A,B,C;  and  so  on.  And  further,  from  §  133,  this 
proposition  can  be  made  good  of  all  universals  when  it  is  known 
of  one :  and  the  same  of  all  particulars. 

*  The  logicians  who  have  recently  introduced  the  distinction  of  extension 
and  comprehension,  have  altogether  missed  this  opposition  of  the  quantities, 
and  have  imagined  that  the  quantities  remain  the  same.  Thus,  according  to 
Sir  W.  Hamilton  'All  X  is  some  Y'  is  a  proposition  of  comprehension,  but 
'  Some  Y  is  all  X'  is  a  proposition  of  extension.  In  this  the  logicians  have 
abandoned  both  Aristotle  and  the  laws  of  thought  from  which  he  drew  the 


few  clear  words  of  his  dictum  :  '  the  genus  is  said  to  be  part  of  the  species ; 
but  in  another  point  of  view  («ja«i)  the  species  is  part  of  the  genus'.  All 
animal  is  in  man,  notion  in  notion  :  all  man  is  in  animal,  class  in  class.  In 
the  first,  all  the  notion  animal  part  of  the  notion  man:  in  the  second,  all 
the  class  man  part  of  the  class  animal.  Here  is  the  opposition  of  the 

215.  It  appears  then  that  the  elements  of  a  tension  (aggregants 
of  an  extent,  components  of  an  intent)  may  be  dismissed  from  the 
term  used  fully,  but  cannot  be  introduced ;   may  be  introduced 
into  a  term  used  vaguely,  but  cannot  be  dismissed.      The  dis- 
missible  is  inadmissible :  the  indismissible  is  admissible. 

216.  Elements  of  either  tension  may,  under  the  limitations  of 
a  rule  to  be  shown,  be  transposed  from  one  term  of  a  proposition 
to  the  other,  either  directly,  or  by  contraversion,  without  either  loss 
or  gain  of  import  to  the  proposition.     Thus  AB)-(Y  is  the  same 
proposition  as  A)-(BY,  and  X))A,B  is  the  same  as  Xa))B.     The 
demonstration  of  this  may  best  be  seen  by  observing  that  every 
universal  is  a  declaration  of  incompossibility,*  and  every  particular 
a  corresponding  declaration  of  compossibility .     Thus  X)-(Y  is  an 
assertion  that  X  and  Y,  as  names  of  one  object,  are  incompossible ; 
and  X(  )Y  that  they  are  compossible.     Again,  X))Y  declares  X 
and  y  to  be  incompossible ;  and  so  on. 

Now  it  will  be  seen  that  AB)-(Y  is  merely  a  statement  that 
the  three  names  A,  B,  Y,  are  incompossible ;  and  so  is  A)'(BY. 
Hence  AB)(Y  and  A>(BY  are  identical.  Similarly  AB(  )Y  and 
A()BY  are  identical,  both  declaring  the  compossibility  of  A,B,  Y: 
or  thus,  if  two  propositions  be  identical,  their  contraries  must  be 
identical.  Hence  we  learn  that  in  Y))a,  b  we  have  YB))a  &c. 
Carrying  this  through  all  transformations,  we  arrive  at  the 
following  rules: — 

1.  In  universal  propositions,  vague  elements  (the  elements  of 
terms  of  either  vague  tension)  are  transposible ;  directly  in  nega- 
tives, by  contraversion  in  affirmatives.      But  full  elements  are 

2.  In  particular  propositions,  full  elements  (the  elements  of 
terms  of  either  full  tension)  are  transposible ;  directly  in  affirm- 
atives, by  contraversion  in  negatives.     But  vague   elements  are 

Thus  in  X))Y,  Y  is  of  vague  extent ;  if  it  be  (A,  B),  its 
aggregant  A  is  transposible,  the  proposition  being  affirmative,  by 
contraversion:  that  is  X))A, B  is  identical  with  Xa))B.  The 
rules  are  for  comparison  and  generalisation,  not  for  use.  Nothing 


can  be  more  evident  than  that  if  every  X  be  either  A  or  B,  every 
•X  which  is  not  A  is  B. 

*  These  good  words  are  Sir  William  Hamilton's  (see  §  14,  note  t),  to 
whom,  in  matters  of  language,  I  am  under  what  he  would  have  called 
obligations  general  and  obligations  special.  His  occasional  writing  of  the 
adjective  after  the  substantive  is  a  useful  revival  of  an  old  practice,  tending 
much  to  clearness.  As  to  my  obligations  special,  he,  finding  the  word 
parenthesis  not  enough  to  erect  his  reader's  hair,  described  my  notation  as 
"horrent  with  mysterious  spiculae".  This  was  the  very  word  I  wanted, 
§  21  :  for  parenthesis  has  come  to  mean,  not  the  punctuating  sign,  but  the 
matter  which  it  includes  :  and  parenthetic  notation  would  have  been 

217.  It  has  in  effect  been  noticed  that  for  every  full  term  in  a 
proposition  a  term  of  as  much  or  less  tension  may  be  substituted ; 
and  for  every  vague  term  a  term  of  as  much  or  more  tension. 
This  is  the  whole  principle  of  onymatic  syllogism,  or  rather  may 
be  made  so:  for  the  varieties  of  principle  upon  which  all  onymatic 
inference  may  be  systematically  introduced  are  numerous.     Thus 
in  X()Y)-(Z,  giving  X(-(Z,  all  we  do  is  to  substitute  for  Y  used 
vaguely  in  extent,  the  as  extensive  or  more  extensive  term  z.     Or 
thus;  —  for  Y  of  full  intent,  we  substitute  the  as  intensive  or  less 
intensive  term  z.     For  Y)*(Z  or  Y))z,  shows  that  z,  if  anything,  is 
of  greater  extent  and  less  intent  than  Y. 

218.  There  are  processes  which  appear  like  transpositions,  but 
are  not  so  in  reality.     Thus  X))PQ  certainly  gives  XP))Q :  here 
is  a  universal  proposition,  in  which  the  element  of  a  full  tension  is 
transposible.     But  not  transposible  within  the  description  in  §  216, 
in  which  it  is  affirmed  that  the  'proposition  after  transposition  is 
identical  with  the  proposition  before  transposition.     This  is  not 
the  case  here;    for  though  X))PQ  gives  XP))Q,  yet  XP))Q 
does  not  give  X))PQ.      Here,  since  X))PQ  gives  X))Q  and 
X))P,  the  term  XP  is  really  X.     And  further,  since  X))PQ 
gives  X))Q  from  whence  XR))Q,  be  R  what  it  may  so  long 
as  XR  has  existence,  the  deduction  of  XP))Q  from  X))PQ  is 
a  case  of  something  different  from  mere  transposition :    for  P, 
in  XP))Q,  may  be  changed  into  anything  else. 

219.  The  dismissal  of  the  elements  of  terms  comes  under 
what  may  be  called  the  decomposition  of  propositions.     When  the 
elements  of  both  terms  are  of  the  full  tension,  the  proposition  is  a 
compound  of  m  x  n  propositions,  if  m  and  n  be  the  numbers  of 
elements  in  the  two  terms.     Thus  A,  B))CD  gives  and  is  given 
by  the  four  propositions  A))C,  A))D,  B))C,  B))D. 


220.  Species,  external,  deficient,  coinadequate 
Dependent,  repugnant,  inessential,  inalternative 

must  carry  the  notion  of  full  extent  and  vague  intent.  For 
example,  the  universe  being  England,  farmer  is  a  deficient  of 
landowner :  of  all  the  class  *  farmer,  no  part  is  identical  with  a 
certain  part  of  the  class  landowner.  To  know  this  by  extent  I 
must  know  the  whole  class  farmer :  but  to  know  it  by  intent,  I 
need  not  know  all  the  attributes  of  the  notion  farmer.  Let  there 
be  but  one  of  these  attributes  which  is  not  an  essential  of  land- 
owner, and  the  proposition  is  established. 

Genus,  complement,  exient,  partient 
Essential,  alternative,  independent,  irrepugnant 

must  carry  the  notion  of  vague  extent  and  full  intent.  The 
symbols  will  here  help  the  memory  of  those  who  have  fully 
connected  them  with  the  words. 

*  The  student  must,  in  any  one  proposition,  be  on  his  guard  against 
thinking  inconsistently  of  class  and  of  attribute.  Either  of  these  modes  of 
thought  may  be  chosen,  but  not  both  together,  unless  the  attribute  be 
made  to  distinguish  the  class,  without  exceptions.  For  a  remarkable 
instance,  take  the  word  gentleman :  what  different  things  people  usually 
mean,  according  as  they  are  speaking  by  notion  of  class  or  of  attribute  ;  the 
common  attribute  excludes  a  percentage  of  the  class,  and  admits  many  who 
are  not  of  the  class.  The  reader  may  be  puzzled  to  make  out  the  text, 
unless  his  character  of  landowner  correspond  to  his  class. 

221.  The  rules  of  §  50-52  must  be  translated  as  follows. 
A  vague  term  in  the  conclusion  takes  extent  or  intent  (scope  or 
force)  as  follows. 

1.  In  universal  syllogisms,  if  one  term  of  conclusion  be  of 
vague  scope  or  force,  it  has  the  scope  or  force  of  the  other ;    if 
both,  one  has  the  scope  or  force  of  the  whole  middle  term,  the 
other  of  its  whole  contrary. 

2.  In  fundamental  particular  syllogisms,  the  vague  term  or 
terms   of  the   conclusion   take  scope  or  force  from  the  vague 

3.  In  strengthened  particular  syllogisms,  the  vague   term  or 
terms  of  conclusion  take  scope  or  force  from  the  whole  middle 
term  or  from  its  whole  contrary,  according  to  which  is  of  full 
scope  or  force  in  both  premises. 

222.  For  example,  (actual)  farming  depends  on  occupation 
of  land  (see  the  caution  in  §  191,  often  wanted  in  reference  to  this 


66  HYPOTHETICAL  SYLLOGISM,  ETC.        [222-225. 

very  instance) ;  and  occupation  of  land  is  an  essential  of  county 
respectability :  therefore  farming  and  such  respectability  are  in- 
alternative.  Here  the  terms  of  conclusion  are  both  of  vague 
intent  or  force,  and  the  middle  term  of  full  intent :  the  force  is 
precisely  so  much  as  is  contained  in  the  notion  of  occupying  land. 
Any  component  either  of  actual  farming  or  of  county  respect- 
ability which  can  be  possessed  by  a  non-occupier  of  land  is  of  no 
import  in  the  conclusion  as  from  the  above  premises. 

Take  the  mathematical  form  of  the  above: — Farmers  are  a 
species  of  occupiers  of  land ;  the  county  respectables  are  a  species 
of  occupiers  ;  whence  the  farmer  is  a  coinadequate  of  the  county 
respectable,  or  both  together  do  not  make  up  the  whole  universe 
(that  is,  as  implied,  the  population  of  the  county).  Here  the  con- 
trary of  the  middle  term,  the  class  of  non-occupiers  of  land,  forms 
the  extent  of  coinadequacy  of  the  terms  of  conclusion  implied  in 
the  premises. 

223.  The  admission  of  complex  terms,  and  of  copular  relations 
more  general  than  the  word  of  identification  is,  enable  us   to 
include  in  common  syllogism  all  the  cases  known  as  hypothetical 
syllogisms,  conditional  syllogisms,  disjunctive  syllogisms,  dilem- 
mas, &c.     I  shall  merely  take  a  few  cases  of  these. 

If  P  be  true,  Q  is  true;  but  P  is  true,  therefore  Q  is 
true.  This  is  an  hypothetical  syllogism,  so  called.  To  reduce  it 
to  a  common  syllogism  between  *  Q',  'true',  and  a  middle  term, 
we  have  'Q'  is  l  a  proposition  true  when  P  is  true';  'A  propo- 
sition true  when  P  is  true '  is  '  true '  (because  P  is  true) ;  there- 
fore Q  is  true.  Many  other  ways  might  be  given.  In  truth, 
though  the  reduction  is  possible,  the  law  of  thought  connecting- 
hypothesis  with  necessary  consequence  is  of  a  character  which 
may  claim  to  stand  before  syllogism,  and  to  be  employed  in  it, 
rather  than  the  converse.  But  the  discussion  of  this  subject  is 
not  for  a  syllabus :  see  §  226.  In  a  similar  way  may  be  treated 
'  If  P  be  true,  Q  is  true ;  Q  is  not  true :  therefore  P  is  not  true '. 

224.  Say  that  P  is  either  A,  or  B,  or  C ;  A  is  not  X,  B  is  not 
X,  C  is  not  X ;  then  P  is  not  X.     This  is  the  syllogism 

tP))A,B,C>(X  giving  P>(X 
a  common  syllogism  with  the  middle  term  an  aggregate. 

225.  Either  P  is  true,  or  Q ;  if  P  be  true,  X  is  true ;  if  Q 
be  true,  Y  is  true ;  therefore  either  X  or  Y  is  true.     The  truth 
is  the  alternative  of  the  truth  of  P  or  Q ;  which  is  the  alternative 

225-228.]  BELIEF. — PROBABILITY.  67 

of  the  truth  of  X  or  Y ;    therefore  the  truth  is  the  truth  of  the 
alternative  of  X  or  Y. 

Various  other  instances  will  be  found  in  my  formal  Logic, 
pp.  115-125. 

226.  In  all  syllogisms  the  existence  of  the  middle  term  is  a 
datum.     If  the  conclusion  be  false,  the  syllogism  being  logically 
valid,  and  the  premises  true  if  the  terms  exist,  then  the  non- 
existence  of  one  of  the  terms  is  the  error.      And  if  the  terms 
which  remain  in  the  conclusion  be  existent,  the  nonexistence  of 
the  middle  term  may  be  inferred.     When  the  syllogism  is  sub- 
jective in  character,  the   transition  into  the  objective  syllogism 
frequently  hinges  on  this  point.      Suppose  success  in  a  certain 
undertaking,  such  success  being  conceivable,  depends  both  upon 
X  and  upon  Z :  then  X  and  Z  are  not  subjectively  repugnant. 
Suppose   that   in   objective   reality  they   are   repugnant :    their 
coexistence  being  a  thing  wholly  unknown  and  incredible.      It 
follows  then  that  success  is  objectively  unattainable;  impossible 
as  things  are,  people  say.     The  metaphysical  premises  X((Y))Z, 
X,  Y,  Z,  being  conceivable,  give  X(  )Z :   and  if  X  and  Z  have 
objective  existence,  and  X)'(Z,  it  follows  that  Y  does  not  exist ; 
for  if  it  did,  the  premises  X((Y))Z  would  give  X()Z.     Suppose 
a  qualification  which  depends  both  upon  natural  talent  and  early 
training;    and  suppose  the  talent  to  be  one  which   cannot   be 
developed  early,  as  things  go ;  then,  as  things  go,  the  qualification 
is  unattainable. 

227.  The  remaining  logical  whole  of  which  we  have  to  con- 
sider the  parts  is  belief.     This  feeling  is  one  the  magnitude  of 
which  ranges  between  two  extremes ;   certainty  for,  such  as  we 
have  as  to  the  proposition  *  Two  and  two  make  four ';  and  cer- 
tainty against,  such  as  we  have  as  to  the  proposition  '  Two  and 
two  make  five '.    The  first  has  the  whole  belief,  or  no  unbelief;  the 
second  has  no  belief,  and  the  whole  unbelief.     These  extremes  are 
represented  by  1  and  0,  on  the  scale  of  belief:   and  would  be 
represented  by  0  and  1,  if  we  chose  (which  is  not  necessary)  to 
have  a  scale  of  unbelief. 

228.  That  which  may  be  or  may  not  be  claims*  a  portion  of 
belief  and  a  portion  of  unbelief:  that  is,  we  partly  believe  in  the 
"  may  "  and  partly  in  the  "  may  not"     Thus  if  an  iirn  contain  1 3 
white  balls  and  7  black  balls,  and  nothing  else,  and  I  am  going  to 
draw  a  ball  without  knowing  which,  and  without  more  belief  in 

68  TESTIMONY  AND  ARGUMENT.  [228-231. 

one  ball  than  in  another ;  then  my  belief  in  the  drawing  of  white 
is  to  my  belief  in  the  drawing  of  black  as  13  to  7,  that  is,  1  re- 
presenting certainty,  I  have  £$  of  belief  in  a  white  ball,  and  -^  in 
a  black  ball.  This  is  usually  expressed  by  saying  that  the  odds 
in  favour  of  a  white  ball  are  13  to  7,  and  the  chances,  or  proba- 
bilities, of  the  drawing  of  white  or  black  ball,  are  ^  and  -/^.  I 
shall  call  13:7  the  ratio  of  belief  in  a  white  ball,  or  of  unbelief  in 
a  black  ball;  and  7:13  the  ratio  of  belief  in  a  black  ball,  or  of 
unbelief  in  a  white  ball ;  and  1 3  and  7  the  favourable  and  unfa- 
vourable terms  for  the  white  ball. 

*  Here,  as  in  all  other  things,  there  are  portions  which  are  too  small  to 
be  of  perceptible  effect.  Csesar  may  not  have  died  in  the  manner  stated :  he 
mat/,  if  there  were  such  a  person,  which  may  not  be  true,  have  been  captured 
by  the  Britons,  and  detained  in  captivity  for  the  rest  of  his  life.  But  the 
received  history  absorbs  so  much  of  our  belief  that  we  have  but  a  mere  atom 
to  divide  among  all  the  different  ways  in  which  that  story  may  be  wrong. 
There  are  two  opposite  fallacious  methods  of  thinking :  first,  the  confusion 
of  high  moral  certainty  with  absolute  knowledge  in  right  of  the  nearness  of 
the  quantities  of  belief  in  the  two ;  secondly,  the  confusion  of  high  moral 
certainty  with  matters  of  practical  uncertainty,  in  right  of  the  want  of 
absolute  knowledge  in  both. 

229.  Referring  to  my  Formal  Logic,  for  full  explanation  on 
the  subject,  I  shall  here  only  digest  a  few  rules  relative  to  the 
measures  of  belief  and  unbelief,  in  questions  especially  relating  to 

230.  Any  alteration  of  our  minds  with  respect  to  belief  or 
unbelief  of  a  proposition  is  derived  from  two  sources, — 

1.  Testimony,  assertion  for  or  against  by  those  of  whose  know- 
ledge we  have  some  opinion.  This,  when  absolutely  unimpeachable, 
is  authority;  though  this  word  is  used  loosely  for  testimony  of 
high  value.     Testimony  speaks  to  the  thing  asserted,  to  its  truth 
or  falsehood ;  it  turns  out  good  if  the  proposition  be  true,  and  bad 
if  the  proposition  be  false. 

2.  Argument,  reasoning  for  or  against,  addressed  to  the  mind 
on  its  own  fprce.     This,  when  absolutely  unimpeachable,  is  de- 
monstration; though  this  word  is  used  loosely  for  reasoning  of 
great  force.     Reasoning  speaks,  not  simply  to  the  truth  or  false- 
hood, but  to  the  truth  as  proved  in  one  particular  way.     If  an 
argument  be  invalid,  it  does  not  follow  that   the  proposition  is 
false,  but  only  that  it  cannot  be  established  in  that  one  way. 

231.  When  the  proposer  of  an  argument  believes  in  its  con- 

231-233.]  TESTIMONY  AND  ARGUMENT.  69 

elusion,  he  is  one  of  the  testimonies  in  favour  of  the  conclusion, 
independently  of  his  argument. 

232.  Among  the  testimonies  to  a  conclusion  must  be  counted 
the  receiver  himself,  whose  initial  state  of  mind  enters  as  the 
testimony*  of  a  witness  into  the  mathematical  formulas,  though  a 
thing  of  a  very  different  kind.     Suppose  that,  all  circumstances 
duly  considered  so  far  as  he  is  able,  the  receiver  begins  with  an 
impression  that  the  proposition  in  hand  has  7  to  3  against  it,  or 
(3  :  7)  is  his  ratio  of  belief  at  the  outset.     Upon  this  belief  the 
future  testimonies  and  arguments  are  to  act :    and  the  mathe- 
matical effect  is  the  same  as  if  the  first  witness  bore  testimony 
(7  :  3)  against  the  proposition. 

*  This  is  the  point  on  which  the  mathematical  study  of  this  theory 
throws  most  light.  Simple  as  the  thing  may  appear,  there  is  not  one  writer 
in  a  thousand  who  seems  to  know  that  the  legitimate  result  of  argument  and 
testimony  depends  upon  the  initial  state  of  the  receiver's  mind.  They 
request  him  to  begin  without  any  bias ;  to  make  himself  something  which 
he  is  not  by  an  act  of  his  own  will.  Judges  request  juries  to  dismiss  all 
that  they  know  about  the  case  beforehand  :  and  this  when  the  juries  know, 
and  the  judges  know  that  they  know  it,  that  the  mere  fact  of  the  prisoner's 
appearance  at  the  bar  is  itself  three  or  four  to  one  in  favour  of  his  guilt. 
Now  the  jury  do  not  dismiss  this  presumption,  because  they  cannot :  and 
they  need  not,  because  the  sound  remedy  against  the  presumption  lurks  in 
their  own  minds,  and  is  ready  to  act.  It  would  not  be  advisable  to  discuss 
in  a  short  note  the  method  in  which  common  honesty  manages  to  hit  the 
truth,  in  spite  of  prepossession.  But  I  may  state  my  conviction  that  if  the 
juryman  were  consciously  to  aim  at  being  somebody  else,  that  is,  a  person 
without  any  preconceived  notion,  he  would  give  a  wrong  verdict  far  more 
often  than  he  does.  I  should  recommend  him  not  to  think  about  himself  at 
all,  but  to  forget  himself  altogether,  or  at  least  not  to  be  active  in  bringing 
himself  before  himself;  and  to  listen  to  the  evidence.  And  further,  to 
remember  that  the  inquiry  does  not  terminate  in  the  jury-box ;  that  the 
trial  of  the  evidence  commences  when  the  jury  retire ;  that  the  evidence  of 
eleven  other  men  to  the  character  of  the  evidence  is  itself  part  of  the 
evidence ;  and  that  the  demand  for  unanimity  on  the  part  of  the  jury  is  the 
expression  of  the  determination  of  the  law  that  the  juryman  shall  be  forced, 
if  needful,  to  take  other  opinions  into  account.  I  trust  this  necessity  for 
unanimity  will  never  be  done  away  with. 

233.  In  assigning  numerical  value  to  degrees  of  belief,  we  are 
supposing  cases  which  are  nearly  as  unusual  in  human  affairs  as 
numerically  definite  propositions  (§  13).     But  by  the  study  of 
accurate  data,  supposed  attainable,  we  analyse  the  sources  of  error 
to  which  our  minds  are  subject  in  the  rough  processes  which  our 
state  of  knowledge  obliges  us  to  use. 


234.  The  method  of  compounding  testimonies  is  by  multiply- 
ing together  all  the  favourable  numbers  for  a  favourable  number, 
and  all  the  unfavourable  numbers  for  an  unfavourable  number. 
For  instance,  a  person  thinks  it  10  to  3  against  an  assertion. 
Two  witnesses  affirm  it,  for  whose  accuracy  it  is  in  his  mind 
7  to  4  and  8  to  3 :   two  witnesses  deny  it,  for  whose  accuracy  it 
is  in  his  mind  11  to  5  and  3  to  1.     What  ought  to  be  his  state  of 
belief  after  the  testimony  ? 

The  several  ratios  for  the  assertion  are 

3:  10,  7:4,  8:3,  5:  11,  1:3 

And  3x7x8x5x1  :  10  x  4  x  3  x  11  x  3,  or  7  :  33  is  the 
ratio  of  belief  as  it  should  be  after  the  whole  testimony  is  taken 
into  account :  or  33  to  7  against  the  assertion. 

235.  When  several  arguments  are  advanced  on  one  side  of  a 
question,  of  which  the  several  chances  of  validity  are  given,  the 
chance  that  the  side  taken  is  proved,  that  is,  that  one  or  more  of 
the  arguments  are  valid,  is  as  follows.     Take  the  product  of  the 
unfavourable  numbers  for  the  unfavourable  number,  and  subtract 
it  from  the  product  of  the  several  totals  for  the  favourable  number. 
Thus  if  three  arguments  be  advanced  on  one  side,  the  ratios  of 
belief  in  which  are  (4  :  3),  (2  :  1),  (3  :  7),  the  unfavourable  num- 
ber  is  3x1x7,  which  subtracted  from  the  product  of  4  +  3, 
2  +  1,  3  +  7,  gives  the  favourable  number.      Hence  (189  :  21) 
or  (9  :  1)  is  the  chance  of  the  side  being  established  by  one  or 
more  of  the  arguments. 

236.  Every  argument,  however  weak,  lends  some  force  to  its 
conclusion :  for  it  may  be  valid,  and  if  invalid  does  not  disprove 
the  conclusion.     But  it  must  be  remembered  that  this  conclusion 
is  modified  by  the  argument  on  the  other  side  which  arises  from 
the  production  of  weak  arguments,  or  none  but  weak  arguments. 
Weak  arguments  from  a  strong   person  themselves  furnish  an 
argument.     If  an  assertion  be  true,  it  is  next  to  certain  that  very 
strong  arguments  exist  for  it ;  if  such  arguments  exist,  it  is  highly 
probable  that  such  and  such  a  person  could  find  them :  but  he 
cannot  find  them ;  whence  there  is  strong  presumption  that  the 
arguments  do  not  exist,  and  from  thence  that  the  assertion  is  not 
true.      This  kind  of  reasoning  really  prevails,  and  leads   to  a 
rational  conclusion  that  the  production  of  none  but  weak  argu- 
ments is  a  strong  presumption  against  the  truth  of  their  con- 
clusion.    But  when  weak  arguments  are  mixed  with  strong  ones, 

236-241.]  CALCULATIC         A     000  094  069 

they  may  rather  tend  to  reinforce  the  conclusion,  though  the 
general  impression  is  that  they  only  weaken  their  stronger 

237.  If  ever  an  argument  be  of  such  nature  that  according  as 
it  is  valid  or  invalid  the  conclusion  is  true  or  false,  that  argument 
is  of  the  nature  of  a  testimony,  and  must  be  combined  with  the 
rest  as  in  §  234. 

238.  When  testimony  and  arguments  on  both  sides  are  to  be 
combined,  the  result  is  obtained    as  follows.     Combine  all  the 
testimony  into  one  result,  as  in  §  234,  all  the  arguments  for  as  in 
§  235,  and  all  the  arguments  against  in  the  same  way.     Then 
form  the  favourable  and  unfavourable  numbers  in  the  ratio  of 
belief  required,  as  follows:  — 

Favourable  number.  Unfavourable  number. 

Multiply  together  Multiply  together 

The  favourable  number  of  the  testi-   |   The   unfavourable   number  of  the 

mony  testimony 

The   unfavourable  number   of  the   ;   The  unfavourable  number  of  the 

argument  against  argument  for 

The  total  of  the  argument  for  i   The  total  of  the  argument  against 

For  instance,  testimony  giving  (7  :  3),  argument  for,  (5:2)  and 
argument  against,  (8  :  1),  the  ratio  of  belief  for  the  truth  of  the 
assertion  should  be  (7  x  1  x  7  :  3  x  2  x  9)  or  (49  :  54),  that  is,  it 
is  54  :  49  against  the  assertion  being  true. 

239.  When  testimony  is  evenly  balanced,  (1:1),  it  may  be 
altogether  omitted.     When  the  arguments  for  and  against  are 
evenly  balanced,  the  arguments  may  be  omitted.      When  the 
arguments  on  both  sides  are  very  strong,  even  though  not  evenly 
balanced,  the  mind  may  be  presumed  unable  to  compare  the  two 
very  small  quantities  which  they  want  of  certain  validity,  and  the 
arguments  may  be  treated  as  evenly  balanced. 

240.  When  no  argument  is  offered  for,  let  (0 : 1)  represent 
the  ratio  of  belief  which  is  to  be  used  in  the  above  rule :  and  the 
same  when  no  argument  is  offered  against. 

241.  When  testimony   is    evenly   balanced,   and    argument 
for    is   (m  :  n),   there    being    no    argument    against,  we   have 
(1  x  1  x  m  +  n  :  1  x  n  x  1),  or  m  +  n  :  n  for  the  truth  of  the 
assertion.     Thus,  on  a  matter  on  which  our  minds  have  no  bias, 
an  argument  which  has  only  an  even  chance  of  validity  gives 
2  to  1  for  the  truth  of  the  conclusion. 

72  TESTIMONY  AND  ARGUMENT.  [242-244. 

242.  Any  one  may  wisely  try  a  few  cases,  setting  down  in 
each,  to  the  best  of  his  judgment,  or  rather  feeling,  his  ratios  of 
belief  as  to  testimony,  argument  for,  argument  against,  and  final 
conclusion.     If  the  last  do  not  agree  with  the  calculation  made 
from  the  first  three,  he  does  not  agree  with  himself.     This  he 
may  very  easily  fail  to  do,  for,  in  such  matters  of  appreciation,  one 
element  may  have  more  than  justice  done  to  it  at  the  expense  of 
the  rest,  on  the  principle  laid  down  in  the  Gospel  of  St.  Matthew, 
xxv.  29. 

243.  The  distinction  of  aggregation  and  composition  occurs  in 
the  two  leading  rules  of  application  of  the  theory  of  probabilities. 
When  events  are  mutually  exclusive,  that  is,  when  only  one  of 
them  can  happen,  the  chance  that  one  or  other  shall  happen  is 
found   from   the   separate   chances   of  happening  by  a  rule  of 
aggregation,  namely,  by  addition.     But  when  events  are  entirely 
independent,  so   that   any  two  or   more   of  them  may  happen 
together,  the  chance  of  all  happening  is  found  by  applying  to 
the  separate  chances  a  rule  of  composition,  namely,  multiplication. 
The  connexion  of  the  formulae  of  probability  with  those  of  logic  in 
general  has  been  most  strikingly  illustrated  by  Professor  Boole,  in 
his  Mathematical  Analysis  of  Logic,  Cambridge,  1847,  8vo.,  and 
subsequently  in  his  Investigation  of  the  Laws  of  Tliought,  London, 
1854,  8vo.     In  these  works  the  author  has  made  it  manifest  that 
the  symbolic  language  of  algebra,  framed  wholly  on  notions  of 
number  and  quantity,  is  adequate,  by  what  is  certainly  not  an 
accident,  to  the  representation  of  all  the  laws  of  thought. 

244.  I  end  with  a  word  on  the  new  symbols  which  I  have 
employed.     Most  writers  on  logic  strongly  object  to  all  symbols 
except  the  venerable  Barbara,  Celarent,  &c.  in  §  109.     I  should 
advise  the  reader  not  to  make  up  his  mind  on  this  point  until  he 
has  well  weighed  two  facts  which  nobody  disputes,  both  separately 
and  in  connexion.    First,  logic  is  the  only  science  which  has  made 
no  progress  since  the  revival  of  letters :  secondly,  logic  is  the  only 
science  which  has  produced  no  growth  of  symbols. 

Erratum.     Page  55,  line  20,  for  will  be  puzzled  read  will  not  be  puzzled. 

Printed  by  G.  BARCLAY,  Castle  St.  Leicester  Sq.